research article on cluster -algebras · 2019. 7. 30. · cluster algebras from riemann surfaces....
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Research ArticleOn Cluster 119862lowast-Algebras
Igor V Nikolaev
Department of Mathematics and Computer Science St Johnrsquos University 8000 Utopia Parkway Queens New York NY 11439 USA
Correspondence should be addressed to Igor V Nikolaev igorvnikolaevgmailcom
Received 4 February 2016 Accepted 18 May 2016
Academic Editor Gelu Popescu
Copyright copy 2016 Igor V Nikolaev This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
We introduce a 119862lowast-algebra A(x 119876) attached to the cluster x and a quiver 119876 If 119876119879is the quiver coming from triangulation 119879 of the
Riemann surface 119878 with a finite number of cusps we prove that the primitive spectrum of A(x 119876119879) times R is homeomorphic to
a generic subset of the Teichmuller space of surface 119878 We conclude with an analog of the Tomita-Takesaki theory and the Connesinvariant 119879(M) for the algebra A(x 119876
119879)
1 Introduction
Cluster algebras of rank 119898 are a class of commutative ringsintroduced by [1] Among these algebras one finds coordinaterings of important algebraic varieties like the Grassman-nians and Schubert varieties cluster algebras appear in theTeichmuller theory [2] Unlike the coordinate rings the set ofgenerators 119909
119894of cluster algebra is usually infinite and defined
by induction from a cluster x = (1199091 119909
119898) and a quiver 119876
see [3] for an excellent survey the cluster algebra is denotedby A(x 119876) Notice that A(x 119876) has an additive structureof countable (unperforated) abelian group with an ordersatisfying the Riesz interpolation property see Remark 5 Inother words the cluster algebraA(x 119876) is a dimension groupby the Effros-Handelman-ShenTheorem [4 Theorem 31]
The subject of our paper is an operator algebra A(x 119876)such that 119870
0(A(x 119876)) cong A(x 119876) here 119870
0(A(x 119876)) is the
dimension group of A(x 119876) and cong is an isomorphism ofthe ordered abelian groups [5 Chapter 7] A(x 119876) is anApproximately Finite 119862lowast-algebra (AF-algebra) given by aBratteli diagram derived explicitly from the pair (x 119876) TheAF-algebras were introduced and studied by [6] we refer toA(x 119876) as a cluster 119862lowast-algebra
An exact definition ofA(x 119876) can be found in Section 24to give an idea recall that the pair (x 119876) is called a seed andthe cluster algebra A(x 119876) is generated by seeds obtainedvia mutation of (x 119876) (and its mutants) in all directions 119896where 1 le 119896 le 119898 [3 p 5] The mutation process canbe described by an oriented regular tree T
119898 the vertices of
T119898correspond to the seeds and the outgoing edges to the
mutations in directions 119896 The quotient B(x 119876) of T119898by a
relation identifying equivalent seeds at the same level of T119898
is a graph with cycles (For a quick example of such a graphsee Figure 3)The cluster119862lowast-algebraA(x 119876) is an AF-algebragiven byB(x 119876) regarded as a Bratteli diagram [6]
Let 119878119892119899
be a Riemann surface of genus 119892 ge 0 with119899 ge 1 cusps such that 2119892 minus 2 + 119899 gt 0 denote by 119879
119892119899cong
R6119892minus6+2119899 the (decorated) Teichmuller space of 119878119892119899 that is a
collection of all Riemann surfaces of genus 119892 with 119899 cuspsendowed with the natural topology [7] In what follows wefocus on the algebras A(x 119876
119892119899) with quivers 119876
119892119899coming
from ideal triangulation of 119878119892119899 the corresponding cluster
algebra A(x 119876119892119899) of rank 119898 = 6119892 minus 6 + 3119899 is related to the
Penner coordinates in 119879119892119899
[2]
Example 1 Let 11987811
be a once-punctured torus The idealtriangulation of 119878
11defines theMarkov quiver (such a quiver
is related to solutions in the integer numbers of the equation1199092
1+ 1199092
2+ 1199092
3= 3119909111990921199093considered by A A Markov hence
the name) 11987611
shown in Figure 1 see [2 Example 46] Thecorresponding cluster 119862lowast-algebra A(x 119876
11) of rank 3 can be
written as
A (x 11987611) cong
M
1198680
(1)
where 1198680is a primitive ideal of an AF-algebra M The
unital AF-algebraM was originally defined by [8 Section 3]
Hindawi Publishing CorporationJournal of Function SpacesVolume 2016 Article ID 9639875 8 pageshttpdxdoiorg10115520169639875
2 Journal of Function Spaces
the genuine notation for such an algebra was M1 because
1198700(M1) = (119872
1 1) fl free one-generator unital ℓ-group
that is finitely piecewise affine linear continuous real-valuedfunctions on [0 1] with integer coefficients M
1was subse-
quently rediscovered after two decades by [9] and denotedbyA The remarkable properties ofM
1include the following
features Every primitive ideal ofM1is essential [10Theorem
42]M1is equipped with a faithful invariant tracial state [11
Theorem 31]The center ofM1coincides with the119862lowast-algebra
119862[0 1] of continuous complex-valued functions on [0 1] [9p 976] There is an affine weak lowast-homeomorphism of thestate space of 119862[0 1] onto the space of tracial states on M
1
[10Theorem45] Any state of119862[0 1]has precisely one tracialextension toM
1[12Theorem 25]The automorphism group
ofM1has precisely two connected components [10Theorem
43]TheGaussmap a Bernoulli shift for continued fractionsis generalized in [12] to the noncommutative framework ofM1 In the light of the original definition of M
1and the
fact that the 1198700-functor preserves exact sequences (see eg
[4 Theorem 31]) the primitive spectrum of M1and its
hull-kernel topology is widely known to the lattice-orderedgroup theorists and the MV-algebraists long ago before thelaborious analysis in [9] where M
1is defined in terms
of the Bratteli diagram We refer the reader to the finalpart of the paper [13] for a general result encompassingthe characterization of the prime spectrum of (119872
1 1) cong
PrimM1 Moreover the AF-algebras 119860
120579introduced by [14]
are precisely the infinite-dimensional simple quotients ofM1 this fact was first proved by [8 Theorem 31(i)] and
rediscovered independently by [9] Summing up the abovethe primitive ideals 119868
120579sub M are indexed by numbers 120579 isin R
if 120579 is irrational the quotient M119868120579cong 119860120579 where 119860
120579is
the Effros-Shen algebra In view of (1) the algebra M is anoncommutative coordinate ring of the Teichmuller space11987911 Moreover there exists an analog of the Tomita-Takesaki
theory of modular automorphisms 120590119905| 119905 isin R for algebra
M see Section 4 such automorphisms correspond to theTeichmuller geodesic flow on 119879
11[15] 120590
119905(119868120579) is an ideal of
M for all 119905 isin R where 1205900(119868120579) = 119868
120579 The quotient algebra
M120590119905(119868120579) can be viewed as a noncommutative coordinate
ring of the Riemann surface 11987811 in particular the pairs (120579 119905)
are coordinates in the space 11987911cong R2 We refer the reader to
[16] for a construction of the corresponding functor
Motivated by Example 1 denote by A(x 119876119892119899) the clus-
ter 119862lowast-algebra corresponding to a quiver 119876119892119899 let 120590
119905
A(x 119876119892119899) rarr A(x 119876
119892119899) be the Tomita-Takesaki flow
on A(x 119876119892119899) see Section 4 for the details Denote by
PrimA(x 119876119892119899) the set of all primitive ideals of A(x 119876
119892119899)
endowed with the Jacobson topology and let 119868120579
isin
PrimA(x 119876119892119899) for a generic value of index 120579 isin R6119892minus7+2119899 Our
main result can be stated as follows
Theorem 2 There exists a homeomorphism
ℎPrimA (x 119876119892119899) timesR
997888rarr 119880 sube 119879119892119899| 119880 is generic
(2)
1
23
Figure 1 The Markov quiver 11987611
given by the formula 120590t(119868120579) 997891rarr 119878119892119899 the set119880 = 119879
119892119899if and only
if 119892 = 119899 = 1 120590119905(119868120579) is an ideal of A(x 119876
119892119899) for all 119905 isin R and
the quotient algebra A(x 119876119892119899)120590119905(119868120579) is a noncommutative
coordinate ring of the Riemann surface 119878119892119899
Remark 3 Theorem 2 is valid for 119899 ge 1 that is the Riemannsurfaceswith at least one cuspThis cannot be improved sincethe cluster structure of algebra A(x 119876
119892119899) comes from the
Ptolemy relations satisfied by the Penner coordinates so farsuch coordinates are available only for the Riemann surfaceswith cusps [7] It is likely that the case 119899 = 0 has also a clusterstructure we refer the reader to [17] where a functor from theRiemann surfaces 119878
1198920to the AF-algebras A(x 119876
1198920)120590119905(119868120579)
was constructed
Remark 4 The braid group 1198612119892+119899
with 119899 isin 1 2 admitsa faithful representation by projections in the algebraA(x 119876
119892119899) such a construction is based on the Birman-
Hilden Theorem for the braid groups This observation andthe well-known Laurent phenomenon in the cluster algebra1198700(A(x 119876
119892119899)) allow generalizing the Jones and HOMFLY
invariants of knots and links to an arbitrary number ofvariables see [18] for the details
The paper is organized as follows We introduce prelimi-nary facts and notation in Section 2 Theorem 2 is proved inSection 3 An analog of the Tomita-Takesaki theory of mod-ular automorphisms and the Connes invariant 119879(A(x 119876
119892119899))
of the cluster 119862lowast-algebra A(x 119876119892119899) is constructed
2 NotationIn this section we introduce notation and briefly review somepreliminary factsThe reader is encouraged to consult [1ndash3 67] for the details
21 Cluster Algebras A cluster algebra A of rank 119898 is asubring of the field Q(119909
1 119909
119898) of rational functions in 119899
variables Such an algebra is defined by a pair (x 119861) wherex = (119909
1 119909
119898) is a cluster of variables and 119861 = (119887
119894119895) is a
skew-symmetric integermatrix the new cluster x1015840 is obtainedfrom x by an excision of the variable 119909
119896and replacing it by a
new variable 1199091015840119896subject to an exchange relation
1199091198961199091015840
119896=
119898
prod
119894=1
119909
max(119887119894119896 0)119894
+
119898
prod
119894=1
119909
max(minus119887119894119896 0)119894
(3)
Journal of Function Spaces 3
Since the entries of matrix 119861 are exponents of the monomialsin cluster variables one gets a new pair (x1015840 1198611015840) where 1198611015840 =(1198871015840
119894119895) is a skew-symmetric matrix with
1198871015840
119894119895=
minus119887119894119895
if 119894 = 119896 or 119895 = 119896
119887119894119895+
1003816100381610038161003816119887119894119896
1003816100381610038161003816119887119896119895+ 119887119894119896
10038161003816100381610038161003816119887119896119895
10038161003816100381610038161003816
2
otherwise(4)
For brevity the pair (x 119861) is called a seed and the seed(x1015840 1198611015840) fl (x1015840 120583
119896(119861)) is obtained from (x 119861) by a mutation
120583119896in the direction 119896 where 1 le 119896 le 119898 120583
119896is involution
that is 1205832119896= 119868119889 The matrix 119861 is called mutation finite if
only finitely many new matrices can be produced from 119861
by repeated matrix mutations The cluster algebra A(x 119861)can be defined as the subring of Q(119909
1 119909
119898) generated by
the union of all cluster variables obtained from the initialseed (x 119861) by mutations of (x 119861) (and its iterations) in allpossible directions We will write T
119898to denote an oriented
tree whose vertices are seeds (x1015840 1198611015840) and 119898 outgoing arrowsin each vertex correspond tomutations 120583
119896of the seed (x1015840 1198611015840)
The Laurent phenomenon proved by [1] says that A(x 119861) subZ[xplusmn1] where Z[xplusmn1] is the ring of the Laurent polynomialsin variables x = (119909
1 119909
119899) in other words each generator
119909119894of algebraA(x 119861) can be written as a Laurent polynomial
in 119899 variables with the integer coefficients
Remark 5 The Laurent phenomenon turns the additivestructure of cluster algebra A(x 119861) into a totally orderedabelian group satisfying the Riesz interpolation property thatis a dimension group [4 Theorem 31] the abelian groupwith order comes from the semigroup of the Laurent poly-nomials with positive coefficients see [19] for the details Abackgroundon the partially and totally ordered unperforatedabelian groups with the Riesz interpolation property can befound in [4]
To deal with mutation formulas (3) and (4) in geometricterms recall that a quiver 119876 is an oriented graph given bythe set of vertices 119876
0and the set of arrows 119876
1 an example of
quiver is given in Figure 1 Let 119896 be a vertex of119876 the mutatedat vertex 119896 quiver 120583
119896(119876) has the same set of vertices as 119876 but
the set of arrows is obtained by the following procedure (i)for each subquiver 119894 rarr 119896 rarr 119895 one adds a new arrow 119894 rarr 119895(ii) one reverses all arrows with source or target 119896 (iii) oneremoves the arrows in a maximal set of pairwise disjoint 2-cycles The reader can verify that if one encodes a quiver 119876with 119899 vertices by a skew-symmetric matrix 119861(119876) = (119887
119894119895)with
119887119894119895equal to the number of arrows from vertex 119894 to vertex 119895
then mutation 120583119896of seed (x 119861) coincides with such of the
corresponding quiver 119876 Thus the cluster algebra A(x 119861) isdefined by a quiver 119876 we will denote such an algebra byA(x 119876)
22 Cluster Algebras from Riemann Surfaces Let 119892 and 119899 beintegers such that 119892 ge 0 119899 ge 1 and 2119892 minus 2 + 119899 gt 0Denote by 119878
119892119899a Riemann surface of genus 119892 with the 119899 cusp
points It is known that the fundamental domain of 119878119892119899
canbe triangulated by 6119892 minus 6 + 3119899 geodesic arcs 120574 such that the
footpoints of each arc at the absolute of Lobachevsky planeH = 119909 + 119894119910 isin C | 119910 gt 0 coincide with a (preimage of) cuspof 119878119892119899 If 119897(120574) is the hyperbolic length of 120574 measured (with a
sign) between two horocycles around the footpoints of 120574 thenwe set 120582(120574) = 119890(12)119897(120574) 120582(120574) are known to satisfy the Ptolemyrelation
120582 (1205741) 120582 (1205742) + 120582 (120574
3) 120582 (1205744) = 120582 (120574
5) 120582 (1205746) (5)
where 1205741 120574
4are pairwise opposite sides and 120574
5 1205746are the
diagonals of a geodesic quadrilateral in HDenote by 119879
119892119899the decorated Teichmuller space of 119878
119892119899
that is the set of all complex surfaces of genus 119892 with 119899 cuspsendowed with the natural topology it is known that 119879
119892119899cong
R6119892minus6+2119899
Theorem 6 (see [7]) The map 120582 on the set of 6119892 minus 6 + 3119899geodesic arcs 120574
119894defining triangulation of 119878
119892119899is a homeomor-
phism with the image 119879119892119899
Remark 7 Notice that among 6119892 minus 6 + 3119899 real numbers 120582(120574119894)
there are only 6119892 minus 6 + 2119899 independent since such numbersmust satisfy 119899 Ptolemy relations (5)
Let119879be triangulation of surface 119878119892119899
by 6119892minus6+3119899 geodesicarcs 120574
119894 consider a skew-symmetric matrix 119861
119879= (119887119894119895) where
119887119894119895is equal to the number of triangles in 119879 with sides 120574
119894
and 120574119895in clockwise order minus the number of triangles in
119879 with sides 120574119894and 120574
119895in the counterclockwise order It is
known that matrix 119861119879is always mutation finite The cluster
algebraA(x 119861119879) of rank 6119892 minus 6 + 3119899 is called associated with
triangulation 119879
Example 8 Let 11987811
be a once-punctured torus of Example 1The triangulation119879 of the fundamental domain (R2minusZ2)Z2of 11987811
is sketched in Figure 2 in the charts R2 and Hrespectively It is easy to see that in this case x = (119909
1 1199092 1199093)
with 1199091= 12057423 1199092= 12057434 and 119909
3= 12057424 where 120574
119894119895denotes a
geodesic arc with the footpoints 119894 and 119895 The Ptolemy relation(5) reduces to 1205822(120574
23) + 1205822
(12057434) = 120582
2
(12057424) thus 119879
11cong R2
The reader is encouraged to verify that matrix 119861119879has the
following form
119861119879= (
0 2 minus2
minus2 0 2
2 minus2 0
) (6)
Theorem 9 (see [2]) The cluster algebra A(x 119861119879) does not
depend on triangulation119879 but only on the surface 119878119892119899 namely
replacement of the geodesic arc 120574119896by a new geodesic arc 1205741015840
119896(a
flip of 120574119896) corresponds to a mutation 120583
119896of the seed (x 119861
119879)
Remark 10 In view of Theorems 6 and 9 A(x 119861119879) corre-
sponds to an algebra of functions on the Teichmuller space119879119892119899 such an algebra is an analog of the coordinate ring of
119879119892119899
23 119862lowast-Algebras A 119862lowast-algebra is an algebra119860 overC with anorm 119886 997891rarr 119886 and involution 119886 997891rarr 119886
lowast such that it is complete
4 Journal of Function Spaces
1 2
23 34 4
R2
1 = infin
H
Figure 2 Triangulation of the Riemann surface 11987811
with respect to the norm and 119886119887 le 119886119887 and 119886lowast119886 = 1198862for all 119886 119887 isin 119860 Any commutative 119862lowast-algebra is isomorphicto the algebra119862
0(119883) of continuous complex-valued functions
on some locally compact Hausdorff space 119883 otherwise 119860represents a noncommutative topological space
An AF-algebra (Approximately Finite 119862lowast-algebra) is
defined to be the norm closure of an ascending sequenceof finite-dimensional 119862lowast-algebras 119872
119899 where 119872
119899is the 119862lowast-
algebra of the 119899times119899matrices with entries inC Here the index119899 = (119899
1 119899
119896) represents the semisimple matrix algebra
119872119899= 1198721198991oplus sdot sdot sdot oplus 119872
119899119896 The ascending sequence mentioned
above can be written as
1198721
1205931
997888rarr 1198722
1205932
997888rarr sdot sdot sdot (7)
where 119872119894are the finite-dimensional 119862lowast-algebras and 120593
119894are
the homomorphisms between such algebras The homomor-phisms 120593
119894can be arranged into a graph as follows Let119872
119894=
1198721198941oplus sdot sdot sdot oplus 119872
119894119896and119872
1198941015840 = 119872
1198941015840
1
oplus sdot sdot sdot oplus 1198721198941015840
119896
be the semisimple119862lowast-algebras and 120593
119894 119872119894rarr 119872
1198941015840 the homomorphism One
has two sets of vertices 1198811198941 119881
119894119896and 119881
1198941015840
1
1198811198941015840
119896
joined by119887119903119904
edges whenever the summand 119872119894119903contains 119887
119903119904copies
of the summand 1198721198941015840
119904
under the embedding 120593119894 As 119894 varies
one obtains an infinite graph called the Bratteli diagram ofthe AF-algebra The matrix 119861 = (119887
119903119904) is known as a partial
multiplicitymatrix an infinite sequence of119861119894defines a unique
AF-algebraLet 120579 isin R119899minus1 recall that by the Jacobi-Perron continued
fraction of vector (1 120579) one understands the limit
(
1
1205791
120579119899minus1
)= lim119896rarrinfin
(
(
0 0 sdot sdot sdot 0 1
1 0 sdot sdot sdot 0 119887(1)
1
0 0 sdot sdot sdot 1 119887(1)
119899minus1
)
)
sdot sdot sdot(
(
0 0 sdot sdot sdot 0 1
1 0 sdot sdot sdot 0 119887(119896)
1
0 0 sdot sdot sdot 1 119887(119896)
119899minus1
)
)
(
0
0
1
)
(8)
where 119887(119895)119894isin Ncup0 see for example [20] the limit converges
for a generic subset of vectors 120579 isin R119899minus1 Notice that 119899 =
2 corresponds to (a matrix form of) the regular continuedfraction of 120579 such a fraction is always convergent Moreover
Figure 3 The Bratteli diagram of Markovrsquos cluster 119862lowast-algebra
the Jacobi-Perron fraction is finite if and only if vector 120579 =
(120579119894) where 120579
119894are rational The AF-algebra119860
120579associated with
the vector (1 120579) is defined by the Bratteli diagram with thepartial multiplicity matrices equal to 119861
119896in the Jacobi-Perron
fraction of (1 120579) in particular if 119899 = 2119860120579coincides with the
Effros-Shen algebra [14]
24 Cluster 119862lowast-Algebras Notice that the mutation tree T119898
of a cluster algebra A(x 119861) has grading by levels that is adistance from the root of T
119898Wewill say that a pair of clusters
x and x1015840 are ℓ-equivalent if
(i) x and x1015840 lie at the same level(ii) x and x1015840 coincide modulo a cyclic permutation of
variables 119909119894
(iii) 119861 = 1198611015840
It is not hard to see that ℓ is an equivalence relation on the setof vertices of graph T
119898
Definition 11 By a cluster 119862lowast-algebra A(x 119861) one under-stands an AF-algebra given by the Bratteli diagram B(x 119861)of the form
B (x 119861) fl T119898mod ℓ (9)
The rank ofA(x 119861) is equal to such of cluster algebraA(x 119861)
Example 12 If 119861119879is matrix (6) of Example 8 then B(x 119861
119879)
is shown in Figure 3 (We refer the reader to Section 4 for aproof) Notice that the graphB(x 119861
119879) is a part of the Bratteli
diagram of the Mundici algebraM compare [10 Figure 1]
Remark 13 It is not hard to see thatB(x 119861) is no longer a treeand B(x 119861) is a finite graph if and only if A(x 119861) is a finitecluster algebra
3 Proof
Let 119898 = 3(2119892 minus 2 + 119899) be the rank of cluster 119862lowast-algebraA(x 119876
119892119899) For the sake of clarity we will consider the case
119898 = 3 and the general case119898 isin 3 6 9 separately(i) LetA(x 119861
119879) be the cluster119862lowast-algebra of rank 3 In this
case 2119892minus2+119899 = 1 and either119892 = 0 and 119899 = 3 or else119892 = 119899 = 1Since 119879
03cong 119901119905 is trivial we are left with 119892 = 119899 = 1 that is
the once-punctured torus 11987811
Repeating the argument of Example 8 we get the seed(x 119861119879) where x = (119909
1 1199092 1199093) and the skew-symmetric
matrix 119861119879is given by formula (6)
Journal of Function Spaces 5
(x1 x21 x23+
x2 x3) (x1 x2 x21 x
22+
x3)( x22 x
23+
x1 x2 x3)
(x1 x2 x3)
(x1 x2 x3)(x1 x2 x3)
(x1 x2 x3)
(x1 x21 + ((x21 + x22) x3)2
x2x21 + x
22
x3)
( x22 + ((x21 + x22) x3)2
x1 x2
x21 + x
22
x3)( x
22 + x2
3
x1 x2((x22 + x
23) x1)2 + x
22
x3)
( x22 + x23
x1((x22 + x
23) x1)2 + x
23
x2 x3)
(((x21 + x23) x2)2 + x
23
x1x21 + x
23
x2 x3) (x1 x
21 + x
23
x2x21 + ((x21 + x
23) x2)2
x3)
Figure 4 The mutation tree
Let us verify that matrix 119861119879is mutation finite indeed for
each 119896 isin 1 2 3 the matrix mutation formula (4) gives us120583119896(119861119879) = minus119861
119879
Therefore the exchange relations (3) do not vary it isverified directly that such relations have the following form
11990911199091015840
1= 1199092
2+ 1199092
3
11990921199091015840
2= 1199092
1+ 1199092
3
11990931199091015840
3= 1199092
1+ 1199092
2
(10)
Consider a mutation tree T3shown in Figure 4 the
vertices of T3correspond to the mutations of cluster x =
(1199091 1199092 1199093) following the exchange rules (10)
The reader is encouraged to verify that modulo a cyclicpermutation of variables 1199091015840
1= 1199092 11990910158402= 1199093 11990910158403= 1199091and
1199091015840
1= 1199093 11990910158402= 1199091 11990910158403= 1199092one obtains (resp) the following
equivalences of clusters
12058313(x) = 120583
21(x)
12058323(x) = 120583
31(x)
(11)
where 120583119894119895(x) fl 120583
119895(120583119894(x)) there are no other cluster equiva-
lences for the vertices of the same level of graph T3
To determine the graph B(x 119861119879) one needs to take the
quotient of T3by the ℓ-equivalence relations (11) since the
pattern repeats for each level of T3 one gets the B(x 119861
119879)
shown in Figure 3 The cluster 119862lowast-algebra A(x 119861119879) is an AF-
algebra with the Bratteli diagramB(x 119861119879)
Notice that the Bratteli diagram B(x 119861119879) of our AF-
algebra A(x 119861119879) and such of the Mundici algebra M are
distinct compare [10 Figure 1] yet there is an obviousinclusion of one diagram into another Namely if one erasesa ldquocamelrsquos backrdquo (ie the two extreme sides of the diagram)in the Bratteli diagram of M then one gets exactly thediagram in Figure 3 Formally if G is the Bratteli diagramof the Mundici algebra M the complement G minus B(x 119861
119879)
is a hereditary Bratteli diagram which gives rise to an ideal1198680subM such that
A (x 119861119879) cong
M
1198680
(12)
see [6 Lemma 32] 1198680is a primitive ideal ibid Theorem 38
(It is interesting to calculate the group 1198700(1198680) in the context
of the work of [13])On the other hand the space PrimM (and hence
PrimA(x 119861119879)) is well understood see for example [13] or
[9 Proposition 7] Namely
Prim(
M
1198680
) = 119868120579| 120579 isin R (13)
where 119868120579sub M is such that M119868
120579cong 119860120579is the Effros-Shen
algebra [14] if 120579 is an irrational number or M119868120579cong 119872119902is
finite-dimensional matrix 119862lowast-algebra (and an extension ofsuch by the 119862lowast-algebra of compact operators) if 120579 = 119901119902 isa rational number (Note that the third series of primitiveideals of [9 Proposition 7] correspond to the ideal 119868
0)
6 Journal of Function Spaces
Moreover given the Jacobson topology on PrimM thereexists a homeomorphism
ℎ Prim(
M
1198680
) 997888rarr R (14)
defined by the formula 119868120579997891rarr 120579 see [9 Corollary 12]
Let 120590119905 M119868
0rarr M119868
0be the Tomita-Takesaki flow
that is a one-parameter automorphism group of M1198680 see
Section 4 Because 119868120579sub M119868
0 the image 120590119905(119868
120579) of 119868
120579is
correctly defined for all 119905 isin R 120590119905(119868120579) is an ideal of M119868
0
but not necessarily primitive Since 120590119905is nothing but (an
algebraic form of) the Teichmuller geodesic flow on 11987911
[15]one concludes that that the family of ideals
120590119905(119868120579) sub
M
1198680
| 119905 isin R 120579 isin R (15)
can be taken for a coordinate system in the space 11987911cong R2
In view of (14) andM1198680cong A(x 119876
11) one gets the required
homeomorphism
ℎ PrimA (x 11987611) timesR 997888rarr 119879
11 (16)
such that the quotient algebra A(x 11987611)120590119905(119868120579) is a noncom-
mutative coordinate ring of the Riemann surface 11987811
Remark 14 The family of algebras A(x 11987611)120590119905(119868120579) | 120579 =
Const 119905 isin R are in general pairwise nonisomorphic (Forotherwise all ideals 120590
119905(119868120579) | 119905 isin R were primitive) However
theGrothendieck semigroups119870+0are isomorphic see [14] the
action of 120590119905is given by the following formula (see Section 4)
119870+
0(
A (x 11987611)
120590119905(119868120579)
) cong 119890119905
(Z + Z120579) (17)
(ii) The general case 119898 = 3119896 = 3(2119892 minus 2 + 119899) is treatedlikewise Notice that if 119889 = 6119892 minus 6 + 2119899 is dimension of thespace119879
119892119899 thenwe have119898minus119889 = 119899 in particular rank119898 of the
cluster 119862lowast-algebra A(x 119876119892119899) determines completely the pair
(119892 119899) provided 119889 is a fixed constant (If 119889 is not fixed thereis only a finite number of different pairs (119892 119899) for given rank119898)
Let (x 119861119879) be the seed given by the cluster x =
(1199091 119909
3119896) and the skew-symmetric matrix 119861
119879 Since
matrix 119861119879comes from triangulation of the Riemann surface
119878119892119899 119861119879ismutation finite see [3 p 18] the exchange relations
(3) take the following form
11990911199091015840
1= 1199092
2+ 1199092
3+ sdot sdot sdot + 119909
2
3119896
11990921199091015840
2= 1199092
1+ 1199092
3+ sdot sdot sdot + 119909
2
3119896
11990931198961199091015840
3119896= 1199092
1+ 1199092
2+ sdot sdot sdot + 119909
2
3119896minus1
(18)
One can construct the mutation tree T3119896
using relations(18) the reader is encouraged to verify that T
3119896is similar
middot middot middot middot middot middot middot middot middot middot middot middot middot middot middot middot middot middot
Figure 5 The Bratteli diagram of a cluster 119862lowast-algebra of rank 6
to the one shown in Figure 4 except for the number of theoutgoing edges at each vertex that is equal to 3119896
A tedious but straightforward calculation shows that theonly equivalent clusters at the same level of T
3119896are the ones
at the extremities of tuples (11990910158401 119909
1015840
3119896) in other words one
gets the following system of equivalences of clusters
12058313119896
(x) = 12058321(x)
12058323119896
(x) = 12058331(x)
1205833119896minus13119896
(x) = 12058331198961
(x)
(19)
where 120583119894119895(x) fl 120583
119895(120583119894(x))
The graph B(x 119861119879) is the quotient of T
3119896by the ℓ-
equivalence relations (19) for 119896 = 2 such a graph is sketchedin Figure 5 A(x 119876
119892119899) is an AF-algebra given by the Bratteli
diagramB(x 119861119879)
Lemma 15 The set
PrimA (x 119876119892119899) = 119868
120579| 120579 isin R
6119892minus7+2119899
is generic (20)
where A(x 119876119892119899)119868120579is an AF-algebra 119860
120579associated with the
convergent Jacobi-Perron continued fraction of vector (1 120579) seeSection 23
Proof We adapt the argument of [9 case 119896 = 1] to thecase 119896 ge 1 Let 119889 = 6119892 minus 6 + 2119899 be dimension of thespace 119879
119892119899 Roughly speaking the Bratteli diagram B(x 119861
119879)
of algebraA(x 119876119892119899) can be cut in two disjoint piecesG
120579and
B(x 119861119879)minusG120579 as it is shown by [9 Figure 7]G
120579is a (finite or
infinite) vertical strip of constant ldquowidthrdquo 119889 where 119889 is equalto the number of vertices cut from each level ofB(x 119861
119879)The
reader is encouraged to verify that G120579is exactly the Bratteli
diagram of the AF-algebra119860120579associated with the convergent
Jacobi-Perron continued fraction of a generic vector (1 120579) seeSection 23
On the other hand the complement B(x 119861119879) minus G
120579is
a hereditary Bratteli diagram which defines an ideal 119868120579of
algebra A(x 119876119892119899) such that
A (x 119876119892119899)
119868120579
= 119860120579 (21)
Journal of Function Spaces 7
see [6 Lemma 32] Moreover 119868120579is a primitive ideal [6
Theorem38] (An extra care is required if 120579 = (120579119894) is a rational
vector the complete argument can be found in [9 pp 980ndash985]) Lemma 15 follows
Lemma 16 The sequence of primitive ideals 119868120579119899
converges to119868120579in the Jacobson topology in PrimA(x 119876
119892119899) if and only if the
sequence 120579119899converges to 120579 in the Euclidean space R6119892minus7+2119899
Proof The proof is a straightforward adaption of the argu-ment in [9 pp 986ndash988] we leave it as an exercise to thereader
Let 120590119905 A(x 119876
119892119899) rarr A(x 119876
119892119899) be the Tomita-
Takesaki flow that is the group 120590119905| 119905 isin R of modular
automorphisms of algebra A(x 119876119892119899) see Section 4 Because
119868120579sub A(x 119876
119892119899) the image 120590119905(119868
120579) of 119868120579is correctly defined
for all 119905 isin R 120590119905(119868120579) is an ideal of A(x 119876
119892119899) but not
necessarily a primitive ideal Since 120590119905is an algebraic form
of the Teichmuller geodesic flow on the space 119879119892119899
[15] oneconcludes that the family of ideals
120590119905(119868120579) sub A (x 119876
119892119899) | 119905 isin R 120579 isin R
6119892minus7+2119899
(22)
can be taken for a coordinate system in the space 119879119892119899
cong
R6119892minus6+2119899 In view of Lemmas 15 and 16 one gets the requiredhomeomorphism
ℎ PrimA (x 119876119892119899) timesR
997888rarr 119880 sube 119879119892119899| 119880 is generic
(23)
such that the quotient algebra 119860120579= A(x 119876
119892119899)120590119905(119868120579) is a
noncommutative coordinate ring of theRiemann surface 119878119892119899
Theorem 2 is proved
4 An Analog of Modular Flow on A(x 119876119892119899)
41 Modular Automorphisms 120590119905| 119905 isin R Recall that the
Ptolemy relations (5) for the Penner coordinates 120582(120574119894) in
the space 119879119892119899
are homogeneous in particular the system119905120582(120574119894) | 119905 isin R of such coordinates will also satisfy the
Ptolemy relations On the other hand for the cluster 119862lowast-algebra A(x 119876
119892119899) the variables 119909
119894= 120582(120574
119894) and one gets an
obvious isomorphism A(x 119876119892119899) cong A(119905x 119876
119892119899) for all 119905 isin R
Since A(119905x 119876119892119899) sube A(x 119876
119892119899) one obtains a one-parameter
group of automorphisms
120590119905 A (x 119876
119892119899) 997888rarr A (x 119876
119892119899) (24)
By analogy with [21] we will call 120590119905a Tomita-Takesaki flow on
the cluster119862lowast-algebraA(x 119876119892119899)The reader is encouraged to
verify that120590119905is an algebraic formof the geodesic flow119879119905 on the
Teichmuller space 119879119892119899 see [15] for an introduction Roughly
speaking such a flow comes from the one-parameter groupof matrices
(
119890119905
0
0 119890minus119905
) (25)
acting on the space of holomorphic quadratic differentials ontheRiemann surface 119878
119892119899 the latter is known to be isomorphic
to the Teichmuller space 119879119892119899
42 Connes Invariant119879(A(x 119876119892119899)) Recall that an analogy of
theConnes invariant 119879(M) for a119862lowast-algebraM endowedwithamodular automorphism group120590
119905is the set119879(M) fl 119905 isin R |
120590119905is inner [21] The group of inner automorphisms of the
space 119879119892119899
and algebra A(x 119876119892119899) is isomorphic to the map-
ping class group mod 119878119892119899
of surface 119878119892119899 The automorphism
120601 isin mod119878119892119899
is called pseudo-Anosov if 120601(F120583) = 120582
120601F120583
where F120583is invariant measured foliation and 120582
120601gt 1 is
a constant called dilatation of 120601 120582120601is always an algebraic
number of the maximal degree 6119892 minus 6 + 2119899 [22] It is knownthat if 120601 isin mod119878
119892119899is pseudo-Anosov then there exists
a trajectory O of the geodesic flow 119879119905 and a point 119878
119892119899isin
119879119892119899 such that the points 119878
119892119899and 120601(119878
119892119899) belong to O [15]
O is called an axis of the pseudo-Anosov automorphism120601 The axis can be used to calculate the Connes invariant119879(A(x 119876
119892119899)) of the cluster 119862lowast-algebra A(x 119876
119892119899) indeed in
view of formula (25) one must solve the following system ofequations
120590119905(119909) = 119890
119905
119909
120601 (119909) = 120582120601119909
(26)
for a point 119909 isin O Thus 120590119905(119909) coincides with the inner
automorphism 120601(119909) if and only if 119905 = log 120582120601 Taking all
pseudo-Anosov automorphisms 120601 isin mod119878119892119899 one gets a
formula for the Connes invariant
119879 (A (x 119876119892119899))
= log 120582120601| 120601 isin mod 119878
119892119899is pseudo-Anosov
(27)
Remark 17 The Connes invariant (27) says that the family ofcluster 119862lowast-algebras A(x 119876
119892119899) is an analog of the type III
120582
factors of von Neumann algebras see [21]
Disclosure
All errors and misconceptions in this paper are solely theauthorrsquos
Competing Interests
The author declares that there are no competing interests inthe paper
Acknowledgments
It is the authorrsquos pleasure to thank Ibrahim Assem andthe SAG group of the Department of Mathematics of theUniversity of Sherbrooke for hospitality and excellent work-ing conditions The author is grateful to Ibrahim AssemThomas Brustle Daniele Mundici Ralf Schiffler and VasilisaShramchenko for an introduction to the wonderland ofcluster algebras and helpful correspondence
8 Journal of Function Spaces
References
[1] S Fomin and A Zelevinsky ldquoCluster algebras I foundationsrdquoJournal of the American Mathematical Society vol 15 no 2 pp497ndash529 2002
[2] S Fomin M Shapiro and D Thurston ldquoCluster algebras andtriangulated surfaces I Cluster complexesrdquo Acta Mathematicavol 201 no 1 pp 83ndash146 2008
[3] L K Williams ldquoCluster algebras an introductionrdquo Bulletin ofthe American Mathematical Society vol 51 no 1 pp 1ndash26 2014
[4] E G Effros Dimensions and Clowast-Algebras vol 46 of CBMSRegional Conference Series in Mathematics Conference Boardof the Mathematical Sciences 1981
[5] B Blackadar K-Theory for Operator Algebras vol 5 MSRIPublications Springer 1986
[6] O Bratteli ldquoInductive limits of finite dimensional Clowast-algebrasrdquoTransactions of the American Mathematical Society vol 171 pp195ndash234 1972
[7] R C Penner ldquoThe decorated Teichmuller space of puncturedsurfacesrdquoCommunications inMathematical Physics vol 113 no2 pp 299ndash339 1987
[8] D Mundici ldquoFarey stellar subdivisions ultrasimplicial groupsand119870
0of AF119862-algebrasrdquoAdvances inMathematics vol 68 no
1 pp 23ndash39 1988[9] F P Boca ldquoAnAF algebra associatedwith the Farey tessellationrdquo
Canadian Journal of Mathematics vol 60 no 5 pp 975ndash10002008
[10] D Mundici ldquoRevisiting the Farey AF-algebrardquoMilan Journal ofMathematics vol 79 no 2 pp 643ndash656 2011
[11] D Mundici ldquoRecognizing the Farey-Stern-Brocot AF algebraDedicated to the memory of Renato Caccioppolirdquo RendicontiLincei-Matematica e Applicazioni vol 20 pp 327ndash338 2009
[12] C Eckhardt ldquoA noncommutative Gauss maprdquo MathematicaScandinavica vol 108 no 2 pp 233ndash250 2011
[13] G Panti ldquoPrime ideals in free 119897 -groups and free vector latticesrdquoJournal of Algebra vol 219 no 1 pp 173ndash200 1999
[14] E G Effros and C L Shen ldquoApproximately finite 119862-algebrasand continued fractionsrdquo Indiana University Mathematics Jour-nal vol 29 no 2 pp 191ndash204 1980
[15] W A Veech ldquoThe Teichmuller geodesic flowrdquo Annals ofMathematics Second Series vol 124 no 3 pp 441ndash530 1986
[16] I VNikolaev ldquoOn aTeichmuller functor between the categoriesof complex tori and the Effros-Shen algebrasrdquoNew York Journalof Mathematics vol 15 pp 125ndash132 2009
[17] I Nikolaev ldquoRiemann surfaces and AF-algebrasrdquo Annals ofFunctional Analysis vol 7 no 2 pp 371ndash380 2016
[18] I Nikolaev ldquoCluster Clowast-algebras and knot polynomialsrdquo httparxivorgabs160301180
[19] I Nikolaev ldquoK-theory of cluster Clowast-algebrasrdquo httparxivorgabs151200276
[20] L Bernstein The Jacobi-Perron AlgorithmmdashIts Theory andApplication vol 207 of Lecture Notes in Mathematics SpringerBerlin Germany 1971
[21] A Connes ldquoVon Neumann algebrasrdquo in Proceedings of theInternational Congress of Mathematicians (ICM rsquo78) pp 97ndash109Helsinki Finland 1978
[22] W P Thurston ldquoOn the geometry and dynamics of diffeo-morphisms of surfacesrdquo Bulletin of the American MathematicalSociety vol 19 no 2 pp 417ndash431 1988
Submit your manuscripts athttpwwwhindawicom
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Journal of Function Spaces
the genuine notation for such an algebra was M1 because
1198700(M1) = (119872
1 1) fl free one-generator unital ℓ-group
that is finitely piecewise affine linear continuous real-valuedfunctions on [0 1] with integer coefficients M
1was subse-
quently rediscovered after two decades by [9] and denotedbyA The remarkable properties ofM
1include the following
features Every primitive ideal ofM1is essential [10Theorem
42]M1is equipped with a faithful invariant tracial state [11
Theorem 31]The center ofM1coincides with the119862lowast-algebra
119862[0 1] of continuous complex-valued functions on [0 1] [9p 976] There is an affine weak lowast-homeomorphism of thestate space of 119862[0 1] onto the space of tracial states on M
1
[10Theorem45] Any state of119862[0 1]has precisely one tracialextension toM
1[12Theorem 25]The automorphism group
ofM1has precisely two connected components [10Theorem
43]TheGaussmap a Bernoulli shift for continued fractionsis generalized in [12] to the noncommutative framework ofM1 In the light of the original definition of M
1and the
fact that the 1198700-functor preserves exact sequences (see eg
[4 Theorem 31]) the primitive spectrum of M1and its
hull-kernel topology is widely known to the lattice-orderedgroup theorists and the MV-algebraists long ago before thelaborious analysis in [9] where M
1is defined in terms
of the Bratteli diagram We refer the reader to the finalpart of the paper [13] for a general result encompassingthe characterization of the prime spectrum of (119872
1 1) cong
PrimM1 Moreover the AF-algebras 119860
120579introduced by [14]
are precisely the infinite-dimensional simple quotients ofM1 this fact was first proved by [8 Theorem 31(i)] and
rediscovered independently by [9] Summing up the abovethe primitive ideals 119868
120579sub M are indexed by numbers 120579 isin R
if 120579 is irrational the quotient M119868120579cong 119860120579 where 119860
120579is
the Effros-Shen algebra In view of (1) the algebra M is anoncommutative coordinate ring of the Teichmuller space11987911 Moreover there exists an analog of the Tomita-Takesaki
theory of modular automorphisms 120590119905| 119905 isin R for algebra
M see Section 4 such automorphisms correspond to theTeichmuller geodesic flow on 119879
11[15] 120590
119905(119868120579) is an ideal of
M for all 119905 isin R where 1205900(119868120579) = 119868
120579 The quotient algebra
M120590119905(119868120579) can be viewed as a noncommutative coordinate
ring of the Riemann surface 11987811 in particular the pairs (120579 119905)
are coordinates in the space 11987911cong R2 We refer the reader to
[16] for a construction of the corresponding functor
Motivated by Example 1 denote by A(x 119876119892119899) the clus-
ter 119862lowast-algebra corresponding to a quiver 119876119892119899 let 120590
119905
A(x 119876119892119899) rarr A(x 119876
119892119899) be the Tomita-Takesaki flow
on A(x 119876119892119899) see Section 4 for the details Denote by
PrimA(x 119876119892119899) the set of all primitive ideals of A(x 119876
119892119899)
endowed with the Jacobson topology and let 119868120579
isin
PrimA(x 119876119892119899) for a generic value of index 120579 isin R6119892minus7+2119899 Our
main result can be stated as follows
Theorem 2 There exists a homeomorphism
ℎPrimA (x 119876119892119899) timesR
997888rarr 119880 sube 119879119892119899| 119880 is generic
(2)
1
23
Figure 1 The Markov quiver 11987611
given by the formula 120590t(119868120579) 997891rarr 119878119892119899 the set119880 = 119879
119892119899if and only
if 119892 = 119899 = 1 120590119905(119868120579) is an ideal of A(x 119876
119892119899) for all 119905 isin R and
the quotient algebra A(x 119876119892119899)120590119905(119868120579) is a noncommutative
coordinate ring of the Riemann surface 119878119892119899
Remark 3 Theorem 2 is valid for 119899 ge 1 that is the Riemannsurfaceswith at least one cuspThis cannot be improved sincethe cluster structure of algebra A(x 119876
119892119899) comes from the
Ptolemy relations satisfied by the Penner coordinates so farsuch coordinates are available only for the Riemann surfaceswith cusps [7] It is likely that the case 119899 = 0 has also a clusterstructure we refer the reader to [17] where a functor from theRiemann surfaces 119878
1198920to the AF-algebras A(x 119876
1198920)120590119905(119868120579)
was constructed
Remark 4 The braid group 1198612119892+119899
with 119899 isin 1 2 admitsa faithful representation by projections in the algebraA(x 119876
119892119899) such a construction is based on the Birman-
Hilden Theorem for the braid groups This observation andthe well-known Laurent phenomenon in the cluster algebra1198700(A(x 119876
119892119899)) allow generalizing the Jones and HOMFLY
invariants of knots and links to an arbitrary number ofvariables see [18] for the details
The paper is organized as follows We introduce prelimi-nary facts and notation in Section 2 Theorem 2 is proved inSection 3 An analog of the Tomita-Takesaki theory of mod-ular automorphisms and the Connes invariant 119879(A(x 119876
119892119899))
of the cluster 119862lowast-algebra A(x 119876119892119899) is constructed
2 NotationIn this section we introduce notation and briefly review somepreliminary factsThe reader is encouraged to consult [1ndash3 67] for the details
21 Cluster Algebras A cluster algebra A of rank 119898 is asubring of the field Q(119909
1 119909
119898) of rational functions in 119899
variables Such an algebra is defined by a pair (x 119861) wherex = (119909
1 119909
119898) is a cluster of variables and 119861 = (119887
119894119895) is a
skew-symmetric integermatrix the new cluster x1015840 is obtainedfrom x by an excision of the variable 119909
119896and replacing it by a
new variable 1199091015840119896subject to an exchange relation
1199091198961199091015840
119896=
119898
prod
119894=1
119909
max(119887119894119896 0)119894
+
119898
prod
119894=1
119909
max(minus119887119894119896 0)119894
(3)
Journal of Function Spaces 3
Since the entries of matrix 119861 are exponents of the monomialsin cluster variables one gets a new pair (x1015840 1198611015840) where 1198611015840 =(1198871015840
119894119895) is a skew-symmetric matrix with
1198871015840
119894119895=
minus119887119894119895
if 119894 = 119896 or 119895 = 119896
119887119894119895+
1003816100381610038161003816119887119894119896
1003816100381610038161003816119887119896119895+ 119887119894119896
10038161003816100381610038161003816119887119896119895
10038161003816100381610038161003816
2
otherwise(4)
For brevity the pair (x 119861) is called a seed and the seed(x1015840 1198611015840) fl (x1015840 120583
119896(119861)) is obtained from (x 119861) by a mutation
120583119896in the direction 119896 where 1 le 119896 le 119898 120583
119896is involution
that is 1205832119896= 119868119889 The matrix 119861 is called mutation finite if
only finitely many new matrices can be produced from 119861
by repeated matrix mutations The cluster algebra A(x 119861)can be defined as the subring of Q(119909
1 119909
119898) generated by
the union of all cluster variables obtained from the initialseed (x 119861) by mutations of (x 119861) (and its iterations) in allpossible directions We will write T
119898to denote an oriented
tree whose vertices are seeds (x1015840 1198611015840) and 119898 outgoing arrowsin each vertex correspond tomutations 120583
119896of the seed (x1015840 1198611015840)
The Laurent phenomenon proved by [1] says that A(x 119861) subZ[xplusmn1] where Z[xplusmn1] is the ring of the Laurent polynomialsin variables x = (119909
1 119909
119899) in other words each generator
119909119894of algebraA(x 119861) can be written as a Laurent polynomial
in 119899 variables with the integer coefficients
Remark 5 The Laurent phenomenon turns the additivestructure of cluster algebra A(x 119861) into a totally orderedabelian group satisfying the Riesz interpolation property thatis a dimension group [4 Theorem 31] the abelian groupwith order comes from the semigroup of the Laurent poly-nomials with positive coefficients see [19] for the details Abackgroundon the partially and totally ordered unperforatedabelian groups with the Riesz interpolation property can befound in [4]
To deal with mutation formulas (3) and (4) in geometricterms recall that a quiver 119876 is an oriented graph given bythe set of vertices 119876
0and the set of arrows 119876
1 an example of
quiver is given in Figure 1 Let 119896 be a vertex of119876 the mutatedat vertex 119896 quiver 120583
119896(119876) has the same set of vertices as 119876 but
the set of arrows is obtained by the following procedure (i)for each subquiver 119894 rarr 119896 rarr 119895 one adds a new arrow 119894 rarr 119895(ii) one reverses all arrows with source or target 119896 (iii) oneremoves the arrows in a maximal set of pairwise disjoint 2-cycles The reader can verify that if one encodes a quiver 119876with 119899 vertices by a skew-symmetric matrix 119861(119876) = (119887
119894119895)with
119887119894119895equal to the number of arrows from vertex 119894 to vertex 119895
then mutation 120583119896of seed (x 119861) coincides with such of the
corresponding quiver 119876 Thus the cluster algebra A(x 119861) isdefined by a quiver 119876 we will denote such an algebra byA(x 119876)
22 Cluster Algebras from Riemann Surfaces Let 119892 and 119899 beintegers such that 119892 ge 0 119899 ge 1 and 2119892 minus 2 + 119899 gt 0Denote by 119878
119892119899a Riemann surface of genus 119892 with the 119899 cusp
points It is known that the fundamental domain of 119878119892119899
canbe triangulated by 6119892 minus 6 + 3119899 geodesic arcs 120574 such that the
footpoints of each arc at the absolute of Lobachevsky planeH = 119909 + 119894119910 isin C | 119910 gt 0 coincide with a (preimage of) cuspof 119878119892119899 If 119897(120574) is the hyperbolic length of 120574 measured (with a
sign) between two horocycles around the footpoints of 120574 thenwe set 120582(120574) = 119890(12)119897(120574) 120582(120574) are known to satisfy the Ptolemyrelation
120582 (1205741) 120582 (1205742) + 120582 (120574
3) 120582 (1205744) = 120582 (120574
5) 120582 (1205746) (5)
where 1205741 120574
4are pairwise opposite sides and 120574
5 1205746are the
diagonals of a geodesic quadrilateral in HDenote by 119879
119892119899the decorated Teichmuller space of 119878
119892119899
that is the set of all complex surfaces of genus 119892 with 119899 cuspsendowed with the natural topology it is known that 119879
119892119899cong
R6119892minus6+2119899
Theorem 6 (see [7]) The map 120582 on the set of 6119892 minus 6 + 3119899geodesic arcs 120574
119894defining triangulation of 119878
119892119899is a homeomor-
phism with the image 119879119892119899
Remark 7 Notice that among 6119892 minus 6 + 3119899 real numbers 120582(120574119894)
there are only 6119892 minus 6 + 2119899 independent since such numbersmust satisfy 119899 Ptolemy relations (5)
Let119879be triangulation of surface 119878119892119899
by 6119892minus6+3119899 geodesicarcs 120574
119894 consider a skew-symmetric matrix 119861
119879= (119887119894119895) where
119887119894119895is equal to the number of triangles in 119879 with sides 120574
119894
and 120574119895in clockwise order minus the number of triangles in
119879 with sides 120574119894and 120574
119895in the counterclockwise order It is
known that matrix 119861119879is always mutation finite The cluster
algebraA(x 119861119879) of rank 6119892 minus 6 + 3119899 is called associated with
triangulation 119879
Example 8 Let 11987811
be a once-punctured torus of Example 1The triangulation119879 of the fundamental domain (R2minusZ2)Z2of 11987811
is sketched in Figure 2 in the charts R2 and Hrespectively It is easy to see that in this case x = (119909
1 1199092 1199093)
with 1199091= 12057423 1199092= 12057434 and 119909
3= 12057424 where 120574
119894119895denotes a
geodesic arc with the footpoints 119894 and 119895 The Ptolemy relation(5) reduces to 1205822(120574
23) + 1205822
(12057434) = 120582
2
(12057424) thus 119879
11cong R2
The reader is encouraged to verify that matrix 119861119879has the
following form
119861119879= (
0 2 minus2
minus2 0 2
2 minus2 0
) (6)
Theorem 9 (see [2]) The cluster algebra A(x 119861119879) does not
depend on triangulation119879 but only on the surface 119878119892119899 namely
replacement of the geodesic arc 120574119896by a new geodesic arc 1205741015840
119896(a
flip of 120574119896) corresponds to a mutation 120583
119896of the seed (x 119861
119879)
Remark 10 In view of Theorems 6 and 9 A(x 119861119879) corre-
sponds to an algebra of functions on the Teichmuller space119879119892119899 such an algebra is an analog of the coordinate ring of
119879119892119899
23 119862lowast-Algebras A 119862lowast-algebra is an algebra119860 overC with anorm 119886 997891rarr 119886 and involution 119886 997891rarr 119886
lowast such that it is complete
4 Journal of Function Spaces
1 2
23 34 4
R2
1 = infin
H
Figure 2 Triangulation of the Riemann surface 11987811
with respect to the norm and 119886119887 le 119886119887 and 119886lowast119886 = 1198862for all 119886 119887 isin 119860 Any commutative 119862lowast-algebra is isomorphicto the algebra119862
0(119883) of continuous complex-valued functions
on some locally compact Hausdorff space 119883 otherwise 119860represents a noncommutative topological space
An AF-algebra (Approximately Finite 119862lowast-algebra) is
defined to be the norm closure of an ascending sequenceof finite-dimensional 119862lowast-algebras 119872
119899 where 119872
119899is the 119862lowast-
algebra of the 119899times119899matrices with entries inC Here the index119899 = (119899
1 119899
119896) represents the semisimple matrix algebra
119872119899= 1198721198991oplus sdot sdot sdot oplus 119872
119899119896 The ascending sequence mentioned
above can be written as
1198721
1205931
997888rarr 1198722
1205932
997888rarr sdot sdot sdot (7)
where 119872119894are the finite-dimensional 119862lowast-algebras and 120593
119894are
the homomorphisms between such algebras The homomor-phisms 120593
119894can be arranged into a graph as follows Let119872
119894=
1198721198941oplus sdot sdot sdot oplus 119872
119894119896and119872
1198941015840 = 119872
1198941015840
1
oplus sdot sdot sdot oplus 1198721198941015840
119896
be the semisimple119862lowast-algebras and 120593
119894 119872119894rarr 119872
1198941015840 the homomorphism One
has two sets of vertices 1198811198941 119881
119894119896and 119881
1198941015840
1
1198811198941015840
119896
joined by119887119903119904
edges whenever the summand 119872119894119903contains 119887
119903119904copies
of the summand 1198721198941015840
119904
under the embedding 120593119894 As 119894 varies
one obtains an infinite graph called the Bratteli diagram ofthe AF-algebra The matrix 119861 = (119887
119903119904) is known as a partial
multiplicitymatrix an infinite sequence of119861119894defines a unique
AF-algebraLet 120579 isin R119899minus1 recall that by the Jacobi-Perron continued
fraction of vector (1 120579) one understands the limit
(
1
1205791
120579119899minus1
)= lim119896rarrinfin
(
(
0 0 sdot sdot sdot 0 1
1 0 sdot sdot sdot 0 119887(1)
1
0 0 sdot sdot sdot 1 119887(1)
119899minus1
)
)
sdot sdot sdot(
(
0 0 sdot sdot sdot 0 1
1 0 sdot sdot sdot 0 119887(119896)
1
0 0 sdot sdot sdot 1 119887(119896)
119899minus1
)
)
(
0
0
1
)
(8)
where 119887(119895)119894isin Ncup0 see for example [20] the limit converges
for a generic subset of vectors 120579 isin R119899minus1 Notice that 119899 =
2 corresponds to (a matrix form of) the regular continuedfraction of 120579 such a fraction is always convergent Moreover
Figure 3 The Bratteli diagram of Markovrsquos cluster 119862lowast-algebra
the Jacobi-Perron fraction is finite if and only if vector 120579 =
(120579119894) where 120579
119894are rational The AF-algebra119860
120579associated with
the vector (1 120579) is defined by the Bratteli diagram with thepartial multiplicity matrices equal to 119861
119896in the Jacobi-Perron
fraction of (1 120579) in particular if 119899 = 2119860120579coincides with the
Effros-Shen algebra [14]
24 Cluster 119862lowast-Algebras Notice that the mutation tree T119898
of a cluster algebra A(x 119861) has grading by levels that is adistance from the root of T
119898Wewill say that a pair of clusters
x and x1015840 are ℓ-equivalent if
(i) x and x1015840 lie at the same level(ii) x and x1015840 coincide modulo a cyclic permutation of
variables 119909119894
(iii) 119861 = 1198611015840
It is not hard to see that ℓ is an equivalence relation on the setof vertices of graph T
119898
Definition 11 By a cluster 119862lowast-algebra A(x 119861) one under-stands an AF-algebra given by the Bratteli diagram B(x 119861)of the form
B (x 119861) fl T119898mod ℓ (9)
The rank ofA(x 119861) is equal to such of cluster algebraA(x 119861)
Example 12 If 119861119879is matrix (6) of Example 8 then B(x 119861
119879)
is shown in Figure 3 (We refer the reader to Section 4 for aproof) Notice that the graphB(x 119861
119879) is a part of the Bratteli
diagram of the Mundici algebraM compare [10 Figure 1]
Remark 13 It is not hard to see thatB(x 119861) is no longer a treeand B(x 119861) is a finite graph if and only if A(x 119861) is a finitecluster algebra
3 Proof
Let 119898 = 3(2119892 minus 2 + 119899) be the rank of cluster 119862lowast-algebraA(x 119876
119892119899) For the sake of clarity we will consider the case
119898 = 3 and the general case119898 isin 3 6 9 separately(i) LetA(x 119861
119879) be the cluster119862lowast-algebra of rank 3 In this
case 2119892minus2+119899 = 1 and either119892 = 0 and 119899 = 3 or else119892 = 119899 = 1Since 119879
03cong 119901119905 is trivial we are left with 119892 = 119899 = 1 that is
the once-punctured torus 11987811
Repeating the argument of Example 8 we get the seed(x 119861119879) where x = (119909
1 1199092 1199093) and the skew-symmetric
matrix 119861119879is given by formula (6)
Journal of Function Spaces 5
(x1 x21 x23+
x2 x3) (x1 x2 x21 x
22+
x3)( x22 x
23+
x1 x2 x3)
(x1 x2 x3)
(x1 x2 x3)(x1 x2 x3)
(x1 x2 x3)
(x1 x21 + ((x21 + x22) x3)2
x2x21 + x
22
x3)
( x22 + ((x21 + x22) x3)2
x1 x2
x21 + x
22
x3)( x
22 + x2
3
x1 x2((x22 + x
23) x1)2 + x
22
x3)
( x22 + x23
x1((x22 + x
23) x1)2 + x
23
x2 x3)
(((x21 + x23) x2)2 + x
23
x1x21 + x
23
x2 x3) (x1 x
21 + x
23
x2x21 + ((x21 + x
23) x2)2
x3)
Figure 4 The mutation tree
Let us verify that matrix 119861119879is mutation finite indeed for
each 119896 isin 1 2 3 the matrix mutation formula (4) gives us120583119896(119861119879) = minus119861
119879
Therefore the exchange relations (3) do not vary it isverified directly that such relations have the following form
11990911199091015840
1= 1199092
2+ 1199092
3
11990921199091015840
2= 1199092
1+ 1199092
3
11990931199091015840
3= 1199092
1+ 1199092
2
(10)
Consider a mutation tree T3shown in Figure 4 the
vertices of T3correspond to the mutations of cluster x =
(1199091 1199092 1199093) following the exchange rules (10)
The reader is encouraged to verify that modulo a cyclicpermutation of variables 1199091015840
1= 1199092 11990910158402= 1199093 11990910158403= 1199091and
1199091015840
1= 1199093 11990910158402= 1199091 11990910158403= 1199092one obtains (resp) the following
equivalences of clusters
12058313(x) = 120583
21(x)
12058323(x) = 120583
31(x)
(11)
where 120583119894119895(x) fl 120583
119895(120583119894(x)) there are no other cluster equiva-
lences for the vertices of the same level of graph T3
To determine the graph B(x 119861119879) one needs to take the
quotient of T3by the ℓ-equivalence relations (11) since the
pattern repeats for each level of T3 one gets the B(x 119861
119879)
shown in Figure 3 The cluster 119862lowast-algebra A(x 119861119879) is an AF-
algebra with the Bratteli diagramB(x 119861119879)
Notice that the Bratteli diagram B(x 119861119879) of our AF-
algebra A(x 119861119879) and such of the Mundici algebra M are
distinct compare [10 Figure 1] yet there is an obviousinclusion of one diagram into another Namely if one erasesa ldquocamelrsquos backrdquo (ie the two extreme sides of the diagram)in the Bratteli diagram of M then one gets exactly thediagram in Figure 3 Formally if G is the Bratteli diagramof the Mundici algebra M the complement G minus B(x 119861
119879)
is a hereditary Bratteli diagram which gives rise to an ideal1198680subM such that
A (x 119861119879) cong
M
1198680
(12)
see [6 Lemma 32] 1198680is a primitive ideal ibid Theorem 38
(It is interesting to calculate the group 1198700(1198680) in the context
of the work of [13])On the other hand the space PrimM (and hence
PrimA(x 119861119879)) is well understood see for example [13] or
[9 Proposition 7] Namely
Prim(
M
1198680
) = 119868120579| 120579 isin R (13)
where 119868120579sub M is such that M119868
120579cong 119860120579is the Effros-Shen
algebra [14] if 120579 is an irrational number or M119868120579cong 119872119902is
finite-dimensional matrix 119862lowast-algebra (and an extension ofsuch by the 119862lowast-algebra of compact operators) if 120579 = 119901119902 isa rational number (Note that the third series of primitiveideals of [9 Proposition 7] correspond to the ideal 119868
0)
6 Journal of Function Spaces
Moreover given the Jacobson topology on PrimM thereexists a homeomorphism
ℎ Prim(
M
1198680
) 997888rarr R (14)
defined by the formula 119868120579997891rarr 120579 see [9 Corollary 12]
Let 120590119905 M119868
0rarr M119868
0be the Tomita-Takesaki flow
that is a one-parameter automorphism group of M1198680 see
Section 4 Because 119868120579sub M119868
0 the image 120590119905(119868
120579) of 119868
120579is
correctly defined for all 119905 isin R 120590119905(119868120579) is an ideal of M119868
0
but not necessarily primitive Since 120590119905is nothing but (an
algebraic form of) the Teichmuller geodesic flow on 11987911
[15]one concludes that that the family of ideals
120590119905(119868120579) sub
M
1198680
| 119905 isin R 120579 isin R (15)
can be taken for a coordinate system in the space 11987911cong R2
In view of (14) andM1198680cong A(x 119876
11) one gets the required
homeomorphism
ℎ PrimA (x 11987611) timesR 997888rarr 119879
11 (16)
such that the quotient algebra A(x 11987611)120590119905(119868120579) is a noncom-
mutative coordinate ring of the Riemann surface 11987811
Remark 14 The family of algebras A(x 11987611)120590119905(119868120579) | 120579 =
Const 119905 isin R are in general pairwise nonisomorphic (Forotherwise all ideals 120590
119905(119868120579) | 119905 isin R were primitive) However
theGrothendieck semigroups119870+0are isomorphic see [14] the
action of 120590119905is given by the following formula (see Section 4)
119870+
0(
A (x 11987611)
120590119905(119868120579)
) cong 119890119905
(Z + Z120579) (17)
(ii) The general case 119898 = 3119896 = 3(2119892 minus 2 + 119899) is treatedlikewise Notice that if 119889 = 6119892 minus 6 + 2119899 is dimension of thespace119879
119892119899 thenwe have119898minus119889 = 119899 in particular rank119898 of the
cluster 119862lowast-algebra A(x 119876119892119899) determines completely the pair
(119892 119899) provided 119889 is a fixed constant (If 119889 is not fixed thereis only a finite number of different pairs (119892 119899) for given rank119898)
Let (x 119861119879) be the seed given by the cluster x =
(1199091 119909
3119896) and the skew-symmetric matrix 119861
119879 Since
matrix 119861119879comes from triangulation of the Riemann surface
119878119892119899 119861119879ismutation finite see [3 p 18] the exchange relations
(3) take the following form
11990911199091015840
1= 1199092
2+ 1199092
3+ sdot sdot sdot + 119909
2
3119896
11990921199091015840
2= 1199092
1+ 1199092
3+ sdot sdot sdot + 119909
2
3119896
11990931198961199091015840
3119896= 1199092
1+ 1199092
2+ sdot sdot sdot + 119909
2
3119896minus1
(18)
One can construct the mutation tree T3119896
using relations(18) the reader is encouraged to verify that T
3119896is similar
middot middot middot middot middot middot middot middot middot middot middot middot middot middot middot middot middot middot
Figure 5 The Bratteli diagram of a cluster 119862lowast-algebra of rank 6
to the one shown in Figure 4 except for the number of theoutgoing edges at each vertex that is equal to 3119896
A tedious but straightforward calculation shows that theonly equivalent clusters at the same level of T
3119896are the ones
at the extremities of tuples (11990910158401 119909
1015840
3119896) in other words one
gets the following system of equivalences of clusters
12058313119896
(x) = 12058321(x)
12058323119896
(x) = 12058331(x)
1205833119896minus13119896
(x) = 12058331198961
(x)
(19)
where 120583119894119895(x) fl 120583
119895(120583119894(x))
The graph B(x 119861119879) is the quotient of T
3119896by the ℓ-
equivalence relations (19) for 119896 = 2 such a graph is sketchedin Figure 5 A(x 119876
119892119899) is an AF-algebra given by the Bratteli
diagramB(x 119861119879)
Lemma 15 The set
PrimA (x 119876119892119899) = 119868
120579| 120579 isin R
6119892minus7+2119899
is generic (20)
where A(x 119876119892119899)119868120579is an AF-algebra 119860
120579associated with the
convergent Jacobi-Perron continued fraction of vector (1 120579) seeSection 23
Proof We adapt the argument of [9 case 119896 = 1] to thecase 119896 ge 1 Let 119889 = 6119892 minus 6 + 2119899 be dimension of thespace 119879
119892119899 Roughly speaking the Bratteli diagram B(x 119861
119879)
of algebraA(x 119876119892119899) can be cut in two disjoint piecesG
120579and
B(x 119861119879)minusG120579 as it is shown by [9 Figure 7]G
120579is a (finite or
infinite) vertical strip of constant ldquowidthrdquo 119889 where 119889 is equalto the number of vertices cut from each level ofB(x 119861
119879)The
reader is encouraged to verify that G120579is exactly the Bratteli
diagram of the AF-algebra119860120579associated with the convergent
Jacobi-Perron continued fraction of a generic vector (1 120579) seeSection 23
On the other hand the complement B(x 119861119879) minus G
120579is
a hereditary Bratteli diagram which defines an ideal 119868120579of
algebra A(x 119876119892119899) such that
A (x 119876119892119899)
119868120579
= 119860120579 (21)
Journal of Function Spaces 7
see [6 Lemma 32] Moreover 119868120579is a primitive ideal [6
Theorem38] (An extra care is required if 120579 = (120579119894) is a rational
vector the complete argument can be found in [9 pp 980ndash985]) Lemma 15 follows
Lemma 16 The sequence of primitive ideals 119868120579119899
converges to119868120579in the Jacobson topology in PrimA(x 119876
119892119899) if and only if the
sequence 120579119899converges to 120579 in the Euclidean space R6119892minus7+2119899
Proof The proof is a straightforward adaption of the argu-ment in [9 pp 986ndash988] we leave it as an exercise to thereader
Let 120590119905 A(x 119876
119892119899) rarr A(x 119876
119892119899) be the Tomita-
Takesaki flow that is the group 120590119905| 119905 isin R of modular
automorphisms of algebra A(x 119876119892119899) see Section 4 Because
119868120579sub A(x 119876
119892119899) the image 120590119905(119868
120579) of 119868120579is correctly defined
for all 119905 isin R 120590119905(119868120579) is an ideal of A(x 119876
119892119899) but not
necessarily a primitive ideal Since 120590119905is an algebraic form
of the Teichmuller geodesic flow on the space 119879119892119899
[15] oneconcludes that the family of ideals
120590119905(119868120579) sub A (x 119876
119892119899) | 119905 isin R 120579 isin R
6119892minus7+2119899
(22)
can be taken for a coordinate system in the space 119879119892119899
cong
R6119892minus6+2119899 In view of Lemmas 15 and 16 one gets the requiredhomeomorphism
ℎ PrimA (x 119876119892119899) timesR
997888rarr 119880 sube 119879119892119899| 119880 is generic
(23)
such that the quotient algebra 119860120579= A(x 119876
119892119899)120590119905(119868120579) is a
noncommutative coordinate ring of theRiemann surface 119878119892119899
Theorem 2 is proved
4 An Analog of Modular Flow on A(x 119876119892119899)
41 Modular Automorphisms 120590119905| 119905 isin R Recall that the
Ptolemy relations (5) for the Penner coordinates 120582(120574119894) in
the space 119879119892119899
are homogeneous in particular the system119905120582(120574119894) | 119905 isin R of such coordinates will also satisfy the
Ptolemy relations On the other hand for the cluster 119862lowast-algebra A(x 119876
119892119899) the variables 119909
119894= 120582(120574
119894) and one gets an
obvious isomorphism A(x 119876119892119899) cong A(119905x 119876
119892119899) for all 119905 isin R
Since A(119905x 119876119892119899) sube A(x 119876
119892119899) one obtains a one-parameter
group of automorphisms
120590119905 A (x 119876
119892119899) 997888rarr A (x 119876
119892119899) (24)
By analogy with [21] we will call 120590119905a Tomita-Takesaki flow on
the cluster119862lowast-algebraA(x 119876119892119899)The reader is encouraged to
verify that120590119905is an algebraic formof the geodesic flow119879119905 on the
Teichmuller space 119879119892119899 see [15] for an introduction Roughly
speaking such a flow comes from the one-parameter groupof matrices
(
119890119905
0
0 119890minus119905
) (25)
acting on the space of holomorphic quadratic differentials ontheRiemann surface 119878
119892119899 the latter is known to be isomorphic
to the Teichmuller space 119879119892119899
42 Connes Invariant119879(A(x 119876119892119899)) Recall that an analogy of
theConnes invariant 119879(M) for a119862lowast-algebraM endowedwithamodular automorphism group120590
119905is the set119879(M) fl 119905 isin R |
120590119905is inner [21] The group of inner automorphisms of the
space 119879119892119899
and algebra A(x 119876119892119899) is isomorphic to the map-
ping class group mod 119878119892119899
of surface 119878119892119899 The automorphism
120601 isin mod119878119892119899
is called pseudo-Anosov if 120601(F120583) = 120582
120601F120583
where F120583is invariant measured foliation and 120582
120601gt 1 is
a constant called dilatation of 120601 120582120601is always an algebraic
number of the maximal degree 6119892 minus 6 + 2119899 [22] It is knownthat if 120601 isin mod119878
119892119899is pseudo-Anosov then there exists
a trajectory O of the geodesic flow 119879119905 and a point 119878
119892119899isin
119879119892119899 such that the points 119878
119892119899and 120601(119878
119892119899) belong to O [15]
O is called an axis of the pseudo-Anosov automorphism120601 The axis can be used to calculate the Connes invariant119879(A(x 119876
119892119899)) of the cluster 119862lowast-algebra A(x 119876
119892119899) indeed in
view of formula (25) one must solve the following system ofequations
120590119905(119909) = 119890
119905
119909
120601 (119909) = 120582120601119909
(26)
for a point 119909 isin O Thus 120590119905(119909) coincides with the inner
automorphism 120601(119909) if and only if 119905 = log 120582120601 Taking all
pseudo-Anosov automorphisms 120601 isin mod119878119892119899 one gets a
formula for the Connes invariant
119879 (A (x 119876119892119899))
= log 120582120601| 120601 isin mod 119878
119892119899is pseudo-Anosov
(27)
Remark 17 The Connes invariant (27) says that the family ofcluster 119862lowast-algebras A(x 119876
119892119899) is an analog of the type III
120582
factors of von Neumann algebras see [21]
Disclosure
All errors and misconceptions in this paper are solely theauthorrsquos
Competing Interests
The author declares that there are no competing interests inthe paper
Acknowledgments
It is the authorrsquos pleasure to thank Ibrahim Assem andthe SAG group of the Department of Mathematics of theUniversity of Sherbrooke for hospitality and excellent work-ing conditions The author is grateful to Ibrahim AssemThomas Brustle Daniele Mundici Ralf Schiffler and VasilisaShramchenko for an introduction to the wonderland ofcluster algebras and helpful correspondence
8 Journal of Function Spaces
References
[1] S Fomin and A Zelevinsky ldquoCluster algebras I foundationsrdquoJournal of the American Mathematical Society vol 15 no 2 pp497ndash529 2002
[2] S Fomin M Shapiro and D Thurston ldquoCluster algebras andtriangulated surfaces I Cluster complexesrdquo Acta Mathematicavol 201 no 1 pp 83ndash146 2008
[3] L K Williams ldquoCluster algebras an introductionrdquo Bulletin ofthe American Mathematical Society vol 51 no 1 pp 1ndash26 2014
[4] E G Effros Dimensions and Clowast-Algebras vol 46 of CBMSRegional Conference Series in Mathematics Conference Boardof the Mathematical Sciences 1981
[5] B Blackadar K-Theory for Operator Algebras vol 5 MSRIPublications Springer 1986
[6] O Bratteli ldquoInductive limits of finite dimensional Clowast-algebrasrdquoTransactions of the American Mathematical Society vol 171 pp195ndash234 1972
[7] R C Penner ldquoThe decorated Teichmuller space of puncturedsurfacesrdquoCommunications inMathematical Physics vol 113 no2 pp 299ndash339 1987
[8] D Mundici ldquoFarey stellar subdivisions ultrasimplicial groupsand119870
0of AF119862-algebrasrdquoAdvances inMathematics vol 68 no
1 pp 23ndash39 1988[9] F P Boca ldquoAnAF algebra associatedwith the Farey tessellationrdquo
Canadian Journal of Mathematics vol 60 no 5 pp 975ndash10002008
[10] D Mundici ldquoRevisiting the Farey AF-algebrardquoMilan Journal ofMathematics vol 79 no 2 pp 643ndash656 2011
[11] D Mundici ldquoRecognizing the Farey-Stern-Brocot AF algebraDedicated to the memory of Renato Caccioppolirdquo RendicontiLincei-Matematica e Applicazioni vol 20 pp 327ndash338 2009
[12] C Eckhardt ldquoA noncommutative Gauss maprdquo MathematicaScandinavica vol 108 no 2 pp 233ndash250 2011
[13] G Panti ldquoPrime ideals in free 119897 -groups and free vector latticesrdquoJournal of Algebra vol 219 no 1 pp 173ndash200 1999
[14] E G Effros and C L Shen ldquoApproximately finite 119862-algebrasand continued fractionsrdquo Indiana University Mathematics Jour-nal vol 29 no 2 pp 191ndash204 1980
[15] W A Veech ldquoThe Teichmuller geodesic flowrdquo Annals ofMathematics Second Series vol 124 no 3 pp 441ndash530 1986
[16] I VNikolaev ldquoOn aTeichmuller functor between the categoriesof complex tori and the Effros-Shen algebrasrdquoNew York Journalof Mathematics vol 15 pp 125ndash132 2009
[17] I Nikolaev ldquoRiemann surfaces and AF-algebrasrdquo Annals ofFunctional Analysis vol 7 no 2 pp 371ndash380 2016
[18] I Nikolaev ldquoCluster Clowast-algebras and knot polynomialsrdquo httparxivorgabs160301180
[19] I Nikolaev ldquoK-theory of cluster Clowast-algebrasrdquo httparxivorgabs151200276
[20] L Bernstein The Jacobi-Perron AlgorithmmdashIts Theory andApplication vol 207 of Lecture Notes in Mathematics SpringerBerlin Germany 1971
[21] A Connes ldquoVon Neumann algebrasrdquo in Proceedings of theInternational Congress of Mathematicians (ICM rsquo78) pp 97ndash109Helsinki Finland 1978
[22] W P Thurston ldquoOn the geometry and dynamics of diffeo-morphisms of surfacesrdquo Bulletin of the American MathematicalSociety vol 19 no 2 pp 417ndash431 1988
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Stochastic AnalysisInternational Journal of
Journal of Function Spaces 3
Since the entries of matrix 119861 are exponents of the monomialsin cluster variables one gets a new pair (x1015840 1198611015840) where 1198611015840 =(1198871015840
119894119895) is a skew-symmetric matrix with
1198871015840
119894119895=
minus119887119894119895
if 119894 = 119896 or 119895 = 119896
119887119894119895+
1003816100381610038161003816119887119894119896
1003816100381610038161003816119887119896119895+ 119887119894119896
10038161003816100381610038161003816119887119896119895
10038161003816100381610038161003816
2
otherwise(4)
For brevity the pair (x 119861) is called a seed and the seed(x1015840 1198611015840) fl (x1015840 120583
119896(119861)) is obtained from (x 119861) by a mutation
120583119896in the direction 119896 where 1 le 119896 le 119898 120583
119896is involution
that is 1205832119896= 119868119889 The matrix 119861 is called mutation finite if
only finitely many new matrices can be produced from 119861
by repeated matrix mutations The cluster algebra A(x 119861)can be defined as the subring of Q(119909
1 119909
119898) generated by
the union of all cluster variables obtained from the initialseed (x 119861) by mutations of (x 119861) (and its iterations) in allpossible directions We will write T
119898to denote an oriented
tree whose vertices are seeds (x1015840 1198611015840) and 119898 outgoing arrowsin each vertex correspond tomutations 120583
119896of the seed (x1015840 1198611015840)
The Laurent phenomenon proved by [1] says that A(x 119861) subZ[xplusmn1] where Z[xplusmn1] is the ring of the Laurent polynomialsin variables x = (119909
1 119909
119899) in other words each generator
119909119894of algebraA(x 119861) can be written as a Laurent polynomial
in 119899 variables with the integer coefficients
Remark 5 The Laurent phenomenon turns the additivestructure of cluster algebra A(x 119861) into a totally orderedabelian group satisfying the Riesz interpolation property thatis a dimension group [4 Theorem 31] the abelian groupwith order comes from the semigroup of the Laurent poly-nomials with positive coefficients see [19] for the details Abackgroundon the partially and totally ordered unperforatedabelian groups with the Riesz interpolation property can befound in [4]
To deal with mutation formulas (3) and (4) in geometricterms recall that a quiver 119876 is an oriented graph given bythe set of vertices 119876
0and the set of arrows 119876
1 an example of
quiver is given in Figure 1 Let 119896 be a vertex of119876 the mutatedat vertex 119896 quiver 120583
119896(119876) has the same set of vertices as 119876 but
the set of arrows is obtained by the following procedure (i)for each subquiver 119894 rarr 119896 rarr 119895 one adds a new arrow 119894 rarr 119895(ii) one reverses all arrows with source or target 119896 (iii) oneremoves the arrows in a maximal set of pairwise disjoint 2-cycles The reader can verify that if one encodes a quiver 119876with 119899 vertices by a skew-symmetric matrix 119861(119876) = (119887
119894119895)with
119887119894119895equal to the number of arrows from vertex 119894 to vertex 119895
then mutation 120583119896of seed (x 119861) coincides with such of the
corresponding quiver 119876 Thus the cluster algebra A(x 119861) isdefined by a quiver 119876 we will denote such an algebra byA(x 119876)
22 Cluster Algebras from Riemann Surfaces Let 119892 and 119899 beintegers such that 119892 ge 0 119899 ge 1 and 2119892 minus 2 + 119899 gt 0Denote by 119878
119892119899a Riemann surface of genus 119892 with the 119899 cusp
points It is known that the fundamental domain of 119878119892119899
canbe triangulated by 6119892 minus 6 + 3119899 geodesic arcs 120574 such that the
footpoints of each arc at the absolute of Lobachevsky planeH = 119909 + 119894119910 isin C | 119910 gt 0 coincide with a (preimage of) cuspof 119878119892119899 If 119897(120574) is the hyperbolic length of 120574 measured (with a
sign) between two horocycles around the footpoints of 120574 thenwe set 120582(120574) = 119890(12)119897(120574) 120582(120574) are known to satisfy the Ptolemyrelation
120582 (1205741) 120582 (1205742) + 120582 (120574
3) 120582 (1205744) = 120582 (120574
5) 120582 (1205746) (5)
where 1205741 120574
4are pairwise opposite sides and 120574
5 1205746are the
diagonals of a geodesic quadrilateral in HDenote by 119879
119892119899the decorated Teichmuller space of 119878
119892119899
that is the set of all complex surfaces of genus 119892 with 119899 cuspsendowed with the natural topology it is known that 119879
119892119899cong
R6119892minus6+2119899
Theorem 6 (see [7]) The map 120582 on the set of 6119892 minus 6 + 3119899geodesic arcs 120574
119894defining triangulation of 119878
119892119899is a homeomor-
phism with the image 119879119892119899
Remark 7 Notice that among 6119892 minus 6 + 3119899 real numbers 120582(120574119894)
there are only 6119892 minus 6 + 2119899 independent since such numbersmust satisfy 119899 Ptolemy relations (5)
Let119879be triangulation of surface 119878119892119899
by 6119892minus6+3119899 geodesicarcs 120574
119894 consider a skew-symmetric matrix 119861
119879= (119887119894119895) where
119887119894119895is equal to the number of triangles in 119879 with sides 120574
119894
and 120574119895in clockwise order minus the number of triangles in
119879 with sides 120574119894and 120574
119895in the counterclockwise order It is
known that matrix 119861119879is always mutation finite The cluster
algebraA(x 119861119879) of rank 6119892 minus 6 + 3119899 is called associated with
triangulation 119879
Example 8 Let 11987811
be a once-punctured torus of Example 1The triangulation119879 of the fundamental domain (R2minusZ2)Z2of 11987811
is sketched in Figure 2 in the charts R2 and Hrespectively It is easy to see that in this case x = (119909
1 1199092 1199093)
with 1199091= 12057423 1199092= 12057434 and 119909
3= 12057424 where 120574
119894119895denotes a
geodesic arc with the footpoints 119894 and 119895 The Ptolemy relation(5) reduces to 1205822(120574
23) + 1205822
(12057434) = 120582
2
(12057424) thus 119879
11cong R2
The reader is encouraged to verify that matrix 119861119879has the
following form
119861119879= (
0 2 minus2
minus2 0 2
2 minus2 0
) (6)
Theorem 9 (see [2]) The cluster algebra A(x 119861119879) does not
depend on triangulation119879 but only on the surface 119878119892119899 namely
replacement of the geodesic arc 120574119896by a new geodesic arc 1205741015840
119896(a
flip of 120574119896) corresponds to a mutation 120583
119896of the seed (x 119861
119879)
Remark 10 In view of Theorems 6 and 9 A(x 119861119879) corre-
sponds to an algebra of functions on the Teichmuller space119879119892119899 such an algebra is an analog of the coordinate ring of
119879119892119899
23 119862lowast-Algebras A 119862lowast-algebra is an algebra119860 overC with anorm 119886 997891rarr 119886 and involution 119886 997891rarr 119886
lowast such that it is complete
4 Journal of Function Spaces
1 2
23 34 4
R2
1 = infin
H
Figure 2 Triangulation of the Riemann surface 11987811
with respect to the norm and 119886119887 le 119886119887 and 119886lowast119886 = 1198862for all 119886 119887 isin 119860 Any commutative 119862lowast-algebra is isomorphicto the algebra119862
0(119883) of continuous complex-valued functions
on some locally compact Hausdorff space 119883 otherwise 119860represents a noncommutative topological space
An AF-algebra (Approximately Finite 119862lowast-algebra) is
defined to be the norm closure of an ascending sequenceof finite-dimensional 119862lowast-algebras 119872
119899 where 119872
119899is the 119862lowast-
algebra of the 119899times119899matrices with entries inC Here the index119899 = (119899
1 119899
119896) represents the semisimple matrix algebra
119872119899= 1198721198991oplus sdot sdot sdot oplus 119872
119899119896 The ascending sequence mentioned
above can be written as
1198721
1205931
997888rarr 1198722
1205932
997888rarr sdot sdot sdot (7)
where 119872119894are the finite-dimensional 119862lowast-algebras and 120593
119894are
the homomorphisms between such algebras The homomor-phisms 120593
119894can be arranged into a graph as follows Let119872
119894=
1198721198941oplus sdot sdot sdot oplus 119872
119894119896and119872
1198941015840 = 119872
1198941015840
1
oplus sdot sdot sdot oplus 1198721198941015840
119896
be the semisimple119862lowast-algebras and 120593
119894 119872119894rarr 119872
1198941015840 the homomorphism One
has two sets of vertices 1198811198941 119881
119894119896and 119881
1198941015840
1
1198811198941015840
119896
joined by119887119903119904
edges whenever the summand 119872119894119903contains 119887
119903119904copies
of the summand 1198721198941015840
119904
under the embedding 120593119894 As 119894 varies
one obtains an infinite graph called the Bratteli diagram ofthe AF-algebra The matrix 119861 = (119887
119903119904) is known as a partial
multiplicitymatrix an infinite sequence of119861119894defines a unique
AF-algebraLet 120579 isin R119899minus1 recall that by the Jacobi-Perron continued
fraction of vector (1 120579) one understands the limit
(
1
1205791
120579119899minus1
)= lim119896rarrinfin
(
(
0 0 sdot sdot sdot 0 1
1 0 sdot sdot sdot 0 119887(1)
1
0 0 sdot sdot sdot 1 119887(1)
119899minus1
)
)
sdot sdot sdot(
(
0 0 sdot sdot sdot 0 1
1 0 sdot sdot sdot 0 119887(119896)
1
0 0 sdot sdot sdot 1 119887(119896)
119899minus1
)
)
(
0
0
1
)
(8)
where 119887(119895)119894isin Ncup0 see for example [20] the limit converges
for a generic subset of vectors 120579 isin R119899minus1 Notice that 119899 =
2 corresponds to (a matrix form of) the regular continuedfraction of 120579 such a fraction is always convergent Moreover
Figure 3 The Bratteli diagram of Markovrsquos cluster 119862lowast-algebra
the Jacobi-Perron fraction is finite if and only if vector 120579 =
(120579119894) where 120579
119894are rational The AF-algebra119860
120579associated with
the vector (1 120579) is defined by the Bratteli diagram with thepartial multiplicity matrices equal to 119861
119896in the Jacobi-Perron
fraction of (1 120579) in particular if 119899 = 2119860120579coincides with the
Effros-Shen algebra [14]
24 Cluster 119862lowast-Algebras Notice that the mutation tree T119898
of a cluster algebra A(x 119861) has grading by levels that is adistance from the root of T
119898Wewill say that a pair of clusters
x and x1015840 are ℓ-equivalent if
(i) x and x1015840 lie at the same level(ii) x and x1015840 coincide modulo a cyclic permutation of
variables 119909119894
(iii) 119861 = 1198611015840
It is not hard to see that ℓ is an equivalence relation on the setof vertices of graph T
119898
Definition 11 By a cluster 119862lowast-algebra A(x 119861) one under-stands an AF-algebra given by the Bratteli diagram B(x 119861)of the form
B (x 119861) fl T119898mod ℓ (9)
The rank ofA(x 119861) is equal to such of cluster algebraA(x 119861)
Example 12 If 119861119879is matrix (6) of Example 8 then B(x 119861
119879)
is shown in Figure 3 (We refer the reader to Section 4 for aproof) Notice that the graphB(x 119861
119879) is a part of the Bratteli
diagram of the Mundici algebraM compare [10 Figure 1]
Remark 13 It is not hard to see thatB(x 119861) is no longer a treeand B(x 119861) is a finite graph if and only if A(x 119861) is a finitecluster algebra
3 Proof
Let 119898 = 3(2119892 minus 2 + 119899) be the rank of cluster 119862lowast-algebraA(x 119876
119892119899) For the sake of clarity we will consider the case
119898 = 3 and the general case119898 isin 3 6 9 separately(i) LetA(x 119861
119879) be the cluster119862lowast-algebra of rank 3 In this
case 2119892minus2+119899 = 1 and either119892 = 0 and 119899 = 3 or else119892 = 119899 = 1Since 119879
03cong 119901119905 is trivial we are left with 119892 = 119899 = 1 that is
the once-punctured torus 11987811
Repeating the argument of Example 8 we get the seed(x 119861119879) where x = (119909
1 1199092 1199093) and the skew-symmetric
matrix 119861119879is given by formula (6)
Journal of Function Spaces 5
(x1 x21 x23+
x2 x3) (x1 x2 x21 x
22+
x3)( x22 x
23+
x1 x2 x3)
(x1 x2 x3)
(x1 x2 x3)(x1 x2 x3)
(x1 x2 x3)
(x1 x21 + ((x21 + x22) x3)2
x2x21 + x
22
x3)
( x22 + ((x21 + x22) x3)2
x1 x2
x21 + x
22
x3)( x
22 + x2
3
x1 x2((x22 + x
23) x1)2 + x
22
x3)
( x22 + x23
x1((x22 + x
23) x1)2 + x
23
x2 x3)
(((x21 + x23) x2)2 + x
23
x1x21 + x
23
x2 x3) (x1 x
21 + x
23
x2x21 + ((x21 + x
23) x2)2
x3)
Figure 4 The mutation tree
Let us verify that matrix 119861119879is mutation finite indeed for
each 119896 isin 1 2 3 the matrix mutation formula (4) gives us120583119896(119861119879) = minus119861
119879
Therefore the exchange relations (3) do not vary it isverified directly that such relations have the following form
11990911199091015840
1= 1199092
2+ 1199092
3
11990921199091015840
2= 1199092
1+ 1199092
3
11990931199091015840
3= 1199092
1+ 1199092
2
(10)
Consider a mutation tree T3shown in Figure 4 the
vertices of T3correspond to the mutations of cluster x =
(1199091 1199092 1199093) following the exchange rules (10)
The reader is encouraged to verify that modulo a cyclicpermutation of variables 1199091015840
1= 1199092 11990910158402= 1199093 11990910158403= 1199091and
1199091015840
1= 1199093 11990910158402= 1199091 11990910158403= 1199092one obtains (resp) the following
equivalences of clusters
12058313(x) = 120583
21(x)
12058323(x) = 120583
31(x)
(11)
where 120583119894119895(x) fl 120583
119895(120583119894(x)) there are no other cluster equiva-
lences for the vertices of the same level of graph T3
To determine the graph B(x 119861119879) one needs to take the
quotient of T3by the ℓ-equivalence relations (11) since the
pattern repeats for each level of T3 one gets the B(x 119861
119879)
shown in Figure 3 The cluster 119862lowast-algebra A(x 119861119879) is an AF-
algebra with the Bratteli diagramB(x 119861119879)
Notice that the Bratteli diagram B(x 119861119879) of our AF-
algebra A(x 119861119879) and such of the Mundici algebra M are
distinct compare [10 Figure 1] yet there is an obviousinclusion of one diagram into another Namely if one erasesa ldquocamelrsquos backrdquo (ie the two extreme sides of the diagram)in the Bratteli diagram of M then one gets exactly thediagram in Figure 3 Formally if G is the Bratteli diagramof the Mundici algebra M the complement G minus B(x 119861
119879)
is a hereditary Bratteli diagram which gives rise to an ideal1198680subM such that
A (x 119861119879) cong
M
1198680
(12)
see [6 Lemma 32] 1198680is a primitive ideal ibid Theorem 38
(It is interesting to calculate the group 1198700(1198680) in the context
of the work of [13])On the other hand the space PrimM (and hence
PrimA(x 119861119879)) is well understood see for example [13] or
[9 Proposition 7] Namely
Prim(
M
1198680
) = 119868120579| 120579 isin R (13)
where 119868120579sub M is such that M119868
120579cong 119860120579is the Effros-Shen
algebra [14] if 120579 is an irrational number or M119868120579cong 119872119902is
finite-dimensional matrix 119862lowast-algebra (and an extension ofsuch by the 119862lowast-algebra of compact operators) if 120579 = 119901119902 isa rational number (Note that the third series of primitiveideals of [9 Proposition 7] correspond to the ideal 119868
0)
6 Journal of Function Spaces
Moreover given the Jacobson topology on PrimM thereexists a homeomorphism
ℎ Prim(
M
1198680
) 997888rarr R (14)
defined by the formula 119868120579997891rarr 120579 see [9 Corollary 12]
Let 120590119905 M119868
0rarr M119868
0be the Tomita-Takesaki flow
that is a one-parameter automorphism group of M1198680 see
Section 4 Because 119868120579sub M119868
0 the image 120590119905(119868
120579) of 119868
120579is
correctly defined for all 119905 isin R 120590119905(119868120579) is an ideal of M119868
0
but not necessarily primitive Since 120590119905is nothing but (an
algebraic form of) the Teichmuller geodesic flow on 11987911
[15]one concludes that that the family of ideals
120590119905(119868120579) sub
M
1198680
| 119905 isin R 120579 isin R (15)
can be taken for a coordinate system in the space 11987911cong R2
In view of (14) andM1198680cong A(x 119876
11) one gets the required
homeomorphism
ℎ PrimA (x 11987611) timesR 997888rarr 119879
11 (16)
such that the quotient algebra A(x 11987611)120590119905(119868120579) is a noncom-
mutative coordinate ring of the Riemann surface 11987811
Remark 14 The family of algebras A(x 11987611)120590119905(119868120579) | 120579 =
Const 119905 isin R are in general pairwise nonisomorphic (Forotherwise all ideals 120590
119905(119868120579) | 119905 isin R were primitive) However
theGrothendieck semigroups119870+0are isomorphic see [14] the
action of 120590119905is given by the following formula (see Section 4)
119870+
0(
A (x 11987611)
120590119905(119868120579)
) cong 119890119905
(Z + Z120579) (17)
(ii) The general case 119898 = 3119896 = 3(2119892 minus 2 + 119899) is treatedlikewise Notice that if 119889 = 6119892 minus 6 + 2119899 is dimension of thespace119879
119892119899 thenwe have119898minus119889 = 119899 in particular rank119898 of the
cluster 119862lowast-algebra A(x 119876119892119899) determines completely the pair
(119892 119899) provided 119889 is a fixed constant (If 119889 is not fixed thereis only a finite number of different pairs (119892 119899) for given rank119898)
Let (x 119861119879) be the seed given by the cluster x =
(1199091 119909
3119896) and the skew-symmetric matrix 119861
119879 Since
matrix 119861119879comes from triangulation of the Riemann surface
119878119892119899 119861119879ismutation finite see [3 p 18] the exchange relations
(3) take the following form
11990911199091015840
1= 1199092
2+ 1199092
3+ sdot sdot sdot + 119909
2
3119896
11990921199091015840
2= 1199092
1+ 1199092
3+ sdot sdot sdot + 119909
2
3119896
11990931198961199091015840
3119896= 1199092
1+ 1199092
2+ sdot sdot sdot + 119909
2
3119896minus1
(18)
One can construct the mutation tree T3119896
using relations(18) the reader is encouraged to verify that T
3119896is similar
middot middot middot middot middot middot middot middot middot middot middot middot middot middot middot middot middot middot
Figure 5 The Bratteli diagram of a cluster 119862lowast-algebra of rank 6
to the one shown in Figure 4 except for the number of theoutgoing edges at each vertex that is equal to 3119896
A tedious but straightforward calculation shows that theonly equivalent clusters at the same level of T
3119896are the ones
at the extremities of tuples (11990910158401 119909
1015840
3119896) in other words one
gets the following system of equivalences of clusters
12058313119896
(x) = 12058321(x)
12058323119896
(x) = 12058331(x)
1205833119896minus13119896
(x) = 12058331198961
(x)
(19)
where 120583119894119895(x) fl 120583
119895(120583119894(x))
The graph B(x 119861119879) is the quotient of T
3119896by the ℓ-
equivalence relations (19) for 119896 = 2 such a graph is sketchedin Figure 5 A(x 119876
119892119899) is an AF-algebra given by the Bratteli
diagramB(x 119861119879)
Lemma 15 The set
PrimA (x 119876119892119899) = 119868
120579| 120579 isin R
6119892minus7+2119899
is generic (20)
where A(x 119876119892119899)119868120579is an AF-algebra 119860
120579associated with the
convergent Jacobi-Perron continued fraction of vector (1 120579) seeSection 23
Proof We adapt the argument of [9 case 119896 = 1] to thecase 119896 ge 1 Let 119889 = 6119892 minus 6 + 2119899 be dimension of thespace 119879
119892119899 Roughly speaking the Bratteli diagram B(x 119861
119879)
of algebraA(x 119876119892119899) can be cut in two disjoint piecesG
120579and
B(x 119861119879)minusG120579 as it is shown by [9 Figure 7]G
120579is a (finite or
infinite) vertical strip of constant ldquowidthrdquo 119889 where 119889 is equalto the number of vertices cut from each level ofB(x 119861
119879)The
reader is encouraged to verify that G120579is exactly the Bratteli
diagram of the AF-algebra119860120579associated with the convergent
Jacobi-Perron continued fraction of a generic vector (1 120579) seeSection 23
On the other hand the complement B(x 119861119879) minus G
120579is
a hereditary Bratteli diagram which defines an ideal 119868120579of
algebra A(x 119876119892119899) such that
A (x 119876119892119899)
119868120579
= 119860120579 (21)
Journal of Function Spaces 7
see [6 Lemma 32] Moreover 119868120579is a primitive ideal [6
Theorem38] (An extra care is required if 120579 = (120579119894) is a rational
vector the complete argument can be found in [9 pp 980ndash985]) Lemma 15 follows
Lemma 16 The sequence of primitive ideals 119868120579119899
converges to119868120579in the Jacobson topology in PrimA(x 119876
119892119899) if and only if the
sequence 120579119899converges to 120579 in the Euclidean space R6119892minus7+2119899
Proof The proof is a straightforward adaption of the argu-ment in [9 pp 986ndash988] we leave it as an exercise to thereader
Let 120590119905 A(x 119876
119892119899) rarr A(x 119876
119892119899) be the Tomita-
Takesaki flow that is the group 120590119905| 119905 isin R of modular
automorphisms of algebra A(x 119876119892119899) see Section 4 Because
119868120579sub A(x 119876
119892119899) the image 120590119905(119868
120579) of 119868120579is correctly defined
for all 119905 isin R 120590119905(119868120579) is an ideal of A(x 119876
119892119899) but not
necessarily a primitive ideal Since 120590119905is an algebraic form
of the Teichmuller geodesic flow on the space 119879119892119899
[15] oneconcludes that the family of ideals
120590119905(119868120579) sub A (x 119876
119892119899) | 119905 isin R 120579 isin R
6119892minus7+2119899
(22)
can be taken for a coordinate system in the space 119879119892119899
cong
R6119892minus6+2119899 In view of Lemmas 15 and 16 one gets the requiredhomeomorphism
ℎ PrimA (x 119876119892119899) timesR
997888rarr 119880 sube 119879119892119899| 119880 is generic
(23)
such that the quotient algebra 119860120579= A(x 119876
119892119899)120590119905(119868120579) is a
noncommutative coordinate ring of theRiemann surface 119878119892119899
Theorem 2 is proved
4 An Analog of Modular Flow on A(x 119876119892119899)
41 Modular Automorphisms 120590119905| 119905 isin R Recall that the
Ptolemy relations (5) for the Penner coordinates 120582(120574119894) in
the space 119879119892119899
are homogeneous in particular the system119905120582(120574119894) | 119905 isin R of such coordinates will also satisfy the
Ptolemy relations On the other hand for the cluster 119862lowast-algebra A(x 119876
119892119899) the variables 119909
119894= 120582(120574
119894) and one gets an
obvious isomorphism A(x 119876119892119899) cong A(119905x 119876
119892119899) for all 119905 isin R
Since A(119905x 119876119892119899) sube A(x 119876
119892119899) one obtains a one-parameter
group of automorphisms
120590119905 A (x 119876
119892119899) 997888rarr A (x 119876
119892119899) (24)
By analogy with [21] we will call 120590119905a Tomita-Takesaki flow on
the cluster119862lowast-algebraA(x 119876119892119899)The reader is encouraged to
verify that120590119905is an algebraic formof the geodesic flow119879119905 on the
Teichmuller space 119879119892119899 see [15] for an introduction Roughly
speaking such a flow comes from the one-parameter groupof matrices
(
119890119905
0
0 119890minus119905
) (25)
acting on the space of holomorphic quadratic differentials ontheRiemann surface 119878
119892119899 the latter is known to be isomorphic
to the Teichmuller space 119879119892119899
42 Connes Invariant119879(A(x 119876119892119899)) Recall that an analogy of
theConnes invariant 119879(M) for a119862lowast-algebraM endowedwithamodular automorphism group120590
119905is the set119879(M) fl 119905 isin R |
120590119905is inner [21] The group of inner automorphisms of the
space 119879119892119899
and algebra A(x 119876119892119899) is isomorphic to the map-
ping class group mod 119878119892119899
of surface 119878119892119899 The automorphism
120601 isin mod119878119892119899
is called pseudo-Anosov if 120601(F120583) = 120582
120601F120583
where F120583is invariant measured foliation and 120582
120601gt 1 is
a constant called dilatation of 120601 120582120601is always an algebraic
number of the maximal degree 6119892 minus 6 + 2119899 [22] It is knownthat if 120601 isin mod119878
119892119899is pseudo-Anosov then there exists
a trajectory O of the geodesic flow 119879119905 and a point 119878
119892119899isin
119879119892119899 such that the points 119878
119892119899and 120601(119878
119892119899) belong to O [15]
O is called an axis of the pseudo-Anosov automorphism120601 The axis can be used to calculate the Connes invariant119879(A(x 119876
119892119899)) of the cluster 119862lowast-algebra A(x 119876
119892119899) indeed in
view of formula (25) one must solve the following system ofequations
120590119905(119909) = 119890
119905
119909
120601 (119909) = 120582120601119909
(26)
for a point 119909 isin O Thus 120590119905(119909) coincides with the inner
automorphism 120601(119909) if and only if 119905 = log 120582120601 Taking all
pseudo-Anosov automorphisms 120601 isin mod119878119892119899 one gets a
formula for the Connes invariant
119879 (A (x 119876119892119899))
= log 120582120601| 120601 isin mod 119878
119892119899is pseudo-Anosov
(27)
Remark 17 The Connes invariant (27) says that the family ofcluster 119862lowast-algebras A(x 119876
119892119899) is an analog of the type III
120582
factors of von Neumann algebras see [21]
Disclosure
All errors and misconceptions in this paper are solely theauthorrsquos
Competing Interests
The author declares that there are no competing interests inthe paper
Acknowledgments
It is the authorrsquos pleasure to thank Ibrahim Assem andthe SAG group of the Department of Mathematics of theUniversity of Sherbrooke for hospitality and excellent work-ing conditions The author is grateful to Ibrahim AssemThomas Brustle Daniele Mundici Ralf Schiffler and VasilisaShramchenko for an introduction to the wonderland ofcluster algebras and helpful correspondence
8 Journal of Function Spaces
References
[1] S Fomin and A Zelevinsky ldquoCluster algebras I foundationsrdquoJournal of the American Mathematical Society vol 15 no 2 pp497ndash529 2002
[2] S Fomin M Shapiro and D Thurston ldquoCluster algebras andtriangulated surfaces I Cluster complexesrdquo Acta Mathematicavol 201 no 1 pp 83ndash146 2008
[3] L K Williams ldquoCluster algebras an introductionrdquo Bulletin ofthe American Mathematical Society vol 51 no 1 pp 1ndash26 2014
[4] E G Effros Dimensions and Clowast-Algebras vol 46 of CBMSRegional Conference Series in Mathematics Conference Boardof the Mathematical Sciences 1981
[5] B Blackadar K-Theory for Operator Algebras vol 5 MSRIPublications Springer 1986
[6] O Bratteli ldquoInductive limits of finite dimensional Clowast-algebrasrdquoTransactions of the American Mathematical Society vol 171 pp195ndash234 1972
[7] R C Penner ldquoThe decorated Teichmuller space of puncturedsurfacesrdquoCommunications inMathematical Physics vol 113 no2 pp 299ndash339 1987
[8] D Mundici ldquoFarey stellar subdivisions ultrasimplicial groupsand119870
0of AF119862-algebrasrdquoAdvances inMathematics vol 68 no
1 pp 23ndash39 1988[9] F P Boca ldquoAnAF algebra associatedwith the Farey tessellationrdquo
Canadian Journal of Mathematics vol 60 no 5 pp 975ndash10002008
[10] D Mundici ldquoRevisiting the Farey AF-algebrardquoMilan Journal ofMathematics vol 79 no 2 pp 643ndash656 2011
[11] D Mundici ldquoRecognizing the Farey-Stern-Brocot AF algebraDedicated to the memory of Renato Caccioppolirdquo RendicontiLincei-Matematica e Applicazioni vol 20 pp 327ndash338 2009
[12] C Eckhardt ldquoA noncommutative Gauss maprdquo MathematicaScandinavica vol 108 no 2 pp 233ndash250 2011
[13] G Panti ldquoPrime ideals in free 119897 -groups and free vector latticesrdquoJournal of Algebra vol 219 no 1 pp 173ndash200 1999
[14] E G Effros and C L Shen ldquoApproximately finite 119862-algebrasand continued fractionsrdquo Indiana University Mathematics Jour-nal vol 29 no 2 pp 191ndash204 1980
[15] W A Veech ldquoThe Teichmuller geodesic flowrdquo Annals ofMathematics Second Series vol 124 no 3 pp 441ndash530 1986
[16] I VNikolaev ldquoOn aTeichmuller functor between the categoriesof complex tori and the Effros-Shen algebrasrdquoNew York Journalof Mathematics vol 15 pp 125ndash132 2009
[17] I Nikolaev ldquoRiemann surfaces and AF-algebrasrdquo Annals ofFunctional Analysis vol 7 no 2 pp 371ndash380 2016
[18] I Nikolaev ldquoCluster Clowast-algebras and knot polynomialsrdquo httparxivorgabs160301180
[19] I Nikolaev ldquoK-theory of cluster Clowast-algebrasrdquo httparxivorgabs151200276
[20] L Bernstein The Jacobi-Perron AlgorithmmdashIts Theory andApplication vol 207 of Lecture Notes in Mathematics SpringerBerlin Germany 1971
[21] A Connes ldquoVon Neumann algebrasrdquo in Proceedings of theInternational Congress of Mathematicians (ICM rsquo78) pp 97ndash109Helsinki Finland 1978
[22] W P Thurston ldquoOn the geometry and dynamics of diffeo-morphisms of surfacesrdquo Bulletin of the American MathematicalSociety vol 19 no 2 pp 417ndash431 1988
Submit your manuscripts athttpwwwhindawicom
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Differential EquationsInternational Journal of
Volume 2014
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Journal of
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Mathematical PhysicsAdvances in
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Journal of Function Spaces
1 2
23 34 4
R2
1 = infin
H
Figure 2 Triangulation of the Riemann surface 11987811
with respect to the norm and 119886119887 le 119886119887 and 119886lowast119886 = 1198862for all 119886 119887 isin 119860 Any commutative 119862lowast-algebra is isomorphicto the algebra119862
0(119883) of continuous complex-valued functions
on some locally compact Hausdorff space 119883 otherwise 119860represents a noncommutative topological space
An AF-algebra (Approximately Finite 119862lowast-algebra) is
defined to be the norm closure of an ascending sequenceof finite-dimensional 119862lowast-algebras 119872
119899 where 119872
119899is the 119862lowast-
algebra of the 119899times119899matrices with entries inC Here the index119899 = (119899
1 119899
119896) represents the semisimple matrix algebra
119872119899= 1198721198991oplus sdot sdot sdot oplus 119872
119899119896 The ascending sequence mentioned
above can be written as
1198721
1205931
997888rarr 1198722
1205932
997888rarr sdot sdot sdot (7)
where 119872119894are the finite-dimensional 119862lowast-algebras and 120593
119894are
the homomorphisms between such algebras The homomor-phisms 120593
119894can be arranged into a graph as follows Let119872
119894=
1198721198941oplus sdot sdot sdot oplus 119872
119894119896and119872
1198941015840 = 119872
1198941015840
1
oplus sdot sdot sdot oplus 1198721198941015840
119896
be the semisimple119862lowast-algebras and 120593
119894 119872119894rarr 119872
1198941015840 the homomorphism One
has two sets of vertices 1198811198941 119881
119894119896and 119881
1198941015840
1
1198811198941015840
119896
joined by119887119903119904
edges whenever the summand 119872119894119903contains 119887
119903119904copies
of the summand 1198721198941015840
119904
under the embedding 120593119894 As 119894 varies
one obtains an infinite graph called the Bratteli diagram ofthe AF-algebra The matrix 119861 = (119887
119903119904) is known as a partial
multiplicitymatrix an infinite sequence of119861119894defines a unique
AF-algebraLet 120579 isin R119899minus1 recall that by the Jacobi-Perron continued
fraction of vector (1 120579) one understands the limit
(
1
1205791
120579119899minus1
)= lim119896rarrinfin
(
(
0 0 sdot sdot sdot 0 1
1 0 sdot sdot sdot 0 119887(1)
1
0 0 sdot sdot sdot 1 119887(1)
119899minus1
)
)
sdot sdot sdot(
(
0 0 sdot sdot sdot 0 1
1 0 sdot sdot sdot 0 119887(119896)
1
0 0 sdot sdot sdot 1 119887(119896)
119899minus1
)
)
(
0
0
1
)
(8)
where 119887(119895)119894isin Ncup0 see for example [20] the limit converges
for a generic subset of vectors 120579 isin R119899minus1 Notice that 119899 =
2 corresponds to (a matrix form of) the regular continuedfraction of 120579 such a fraction is always convergent Moreover
Figure 3 The Bratteli diagram of Markovrsquos cluster 119862lowast-algebra
the Jacobi-Perron fraction is finite if and only if vector 120579 =
(120579119894) where 120579
119894are rational The AF-algebra119860
120579associated with
the vector (1 120579) is defined by the Bratteli diagram with thepartial multiplicity matrices equal to 119861
119896in the Jacobi-Perron
fraction of (1 120579) in particular if 119899 = 2119860120579coincides with the
Effros-Shen algebra [14]
24 Cluster 119862lowast-Algebras Notice that the mutation tree T119898
of a cluster algebra A(x 119861) has grading by levels that is adistance from the root of T
119898Wewill say that a pair of clusters
x and x1015840 are ℓ-equivalent if
(i) x and x1015840 lie at the same level(ii) x and x1015840 coincide modulo a cyclic permutation of
variables 119909119894
(iii) 119861 = 1198611015840
It is not hard to see that ℓ is an equivalence relation on the setof vertices of graph T
119898
Definition 11 By a cluster 119862lowast-algebra A(x 119861) one under-stands an AF-algebra given by the Bratteli diagram B(x 119861)of the form
B (x 119861) fl T119898mod ℓ (9)
The rank ofA(x 119861) is equal to such of cluster algebraA(x 119861)
Example 12 If 119861119879is matrix (6) of Example 8 then B(x 119861
119879)
is shown in Figure 3 (We refer the reader to Section 4 for aproof) Notice that the graphB(x 119861
119879) is a part of the Bratteli
diagram of the Mundici algebraM compare [10 Figure 1]
Remark 13 It is not hard to see thatB(x 119861) is no longer a treeand B(x 119861) is a finite graph if and only if A(x 119861) is a finitecluster algebra
3 Proof
Let 119898 = 3(2119892 minus 2 + 119899) be the rank of cluster 119862lowast-algebraA(x 119876
119892119899) For the sake of clarity we will consider the case
119898 = 3 and the general case119898 isin 3 6 9 separately(i) LetA(x 119861
119879) be the cluster119862lowast-algebra of rank 3 In this
case 2119892minus2+119899 = 1 and either119892 = 0 and 119899 = 3 or else119892 = 119899 = 1Since 119879
03cong 119901119905 is trivial we are left with 119892 = 119899 = 1 that is
the once-punctured torus 11987811
Repeating the argument of Example 8 we get the seed(x 119861119879) where x = (119909
1 1199092 1199093) and the skew-symmetric
matrix 119861119879is given by formula (6)
Journal of Function Spaces 5
(x1 x21 x23+
x2 x3) (x1 x2 x21 x
22+
x3)( x22 x
23+
x1 x2 x3)
(x1 x2 x3)
(x1 x2 x3)(x1 x2 x3)
(x1 x2 x3)
(x1 x21 + ((x21 + x22) x3)2
x2x21 + x
22
x3)
( x22 + ((x21 + x22) x3)2
x1 x2
x21 + x
22
x3)( x
22 + x2
3
x1 x2((x22 + x
23) x1)2 + x
22
x3)
( x22 + x23
x1((x22 + x
23) x1)2 + x
23
x2 x3)
(((x21 + x23) x2)2 + x
23
x1x21 + x
23
x2 x3) (x1 x
21 + x
23
x2x21 + ((x21 + x
23) x2)2
x3)
Figure 4 The mutation tree
Let us verify that matrix 119861119879is mutation finite indeed for
each 119896 isin 1 2 3 the matrix mutation formula (4) gives us120583119896(119861119879) = minus119861
119879
Therefore the exchange relations (3) do not vary it isverified directly that such relations have the following form
11990911199091015840
1= 1199092
2+ 1199092
3
11990921199091015840
2= 1199092
1+ 1199092
3
11990931199091015840
3= 1199092
1+ 1199092
2
(10)
Consider a mutation tree T3shown in Figure 4 the
vertices of T3correspond to the mutations of cluster x =
(1199091 1199092 1199093) following the exchange rules (10)
The reader is encouraged to verify that modulo a cyclicpermutation of variables 1199091015840
1= 1199092 11990910158402= 1199093 11990910158403= 1199091and
1199091015840
1= 1199093 11990910158402= 1199091 11990910158403= 1199092one obtains (resp) the following
equivalences of clusters
12058313(x) = 120583
21(x)
12058323(x) = 120583
31(x)
(11)
where 120583119894119895(x) fl 120583
119895(120583119894(x)) there are no other cluster equiva-
lences for the vertices of the same level of graph T3
To determine the graph B(x 119861119879) one needs to take the
quotient of T3by the ℓ-equivalence relations (11) since the
pattern repeats for each level of T3 one gets the B(x 119861
119879)
shown in Figure 3 The cluster 119862lowast-algebra A(x 119861119879) is an AF-
algebra with the Bratteli diagramB(x 119861119879)
Notice that the Bratteli diagram B(x 119861119879) of our AF-
algebra A(x 119861119879) and such of the Mundici algebra M are
distinct compare [10 Figure 1] yet there is an obviousinclusion of one diagram into another Namely if one erasesa ldquocamelrsquos backrdquo (ie the two extreme sides of the diagram)in the Bratteli diagram of M then one gets exactly thediagram in Figure 3 Formally if G is the Bratteli diagramof the Mundici algebra M the complement G minus B(x 119861
119879)
is a hereditary Bratteli diagram which gives rise to an ideal1198680subM such that
A (x 119861119879) cong
M
1198680
(12)
see [6 Lemma 32] 1198680is a primitive ideal ibid Theorem 38
(It is interesting to calculate the group 1198700(1198680) in the context
of the work of [13])On the other hand the space PrimM (and hence
PrimA(x 119861119879)) is well understood see for example [13] or
[9 Proposition 7] Namely
Prim(
M
1198680
) = 119868120579| 120579 isin R (13)
where 119868120579sub M is such that M119868
120579cong 119860120579is the Effros-Shen
algebra [14] if 120579 is an irrational number or M119868120579cong 119872119902is
finite-dimensional matrix 119862lowast-algebra (and an extension ofsuch by the 119862lowast-algebra of compact operators) if 120579 = 119901119902 isa rational number (Note that the third series of primitiveideals of [9 Proposition 7] correspond to the ideal 119868
0)
6 Journal of Function Spaces
Moreover given the Jacobson topology on PrimM thereexists a homeomorphism
ℎ Prim(
M
1198680
) 997888rarr R (14)
defined by the formula 119868120579997891rarr 120579 see [9 Corollary 12]
Let 120590119905 M119868
0rarr M119868
0be the Tomita-Takesaki flow
that is a one-parameter automorphism group of M1198680 see
Section 4 Because 119868120579sub M119868
0 the image 120590119905(119868
120579) of 119868
120579is
correctly defined for all 119905 isin R 120590119905(119868120579) is an ideal of M119868
0
but not necessarily primitive Since 120590119905is nothing but (an
algebraic form of) the Teichmuller geodesic flow on 11987911
[15]one concludes that that the family of ideals
120590119905(119868120579) sub
M
1198680
| 119905 isin R 120579 isin R (15)
can be taken for a coordinate system in the space 11987911cong R2
In view of (14) andM1198680cong A(x 119876
11) one gets the required
homeomorphism
ℎ PrimA (x 11987611) timesR 997888rarr 119879
11 (16)
such that the quotient algebra A(x 11987611)120590119905(119868120579) is a noncom-
mutative coordinate ring of the Riemann surface 11987811
Remark 14 The family of algebras A(x 11987611)120590119905(119868120579) | 120579 =
Const 119905 isin R are in general pairwise nonisomorphic (Forotherwise all ideals 120590
119905(119868120579) | 119905 isin R were primitive) However
theGrothendieck semigroups119870+0are isomorphic see [14] the
action of 120590119905is given by the following formula (see Section 4)
119870+
0(
A (x 11987611)
120590119905(119868120579)
) cong 119890119905
(Z + Z120579) (17)
(ii) The general case 119898 = 3119896 = 3(2119892 minus 2 + 119899) is treatedlikewise Notice that if 119889 = 6119892 minus 6 + 2119899 is dimension of thespace119879
119892119899 thenwe have119898minus119889 = 119899 in particular rank119898 of the
cluster 119862lowast-algebra A(x 119876119892119899) determines completely the pair
(119892 119899) provided 119889 is a fixed constant (If 119889 is not fixed thereis only a finite number of different pairs (119892 119899) for given rank119898)
Let (x 119861119879) be the seed given by the cluster x =
(1199091 119909
3119896) and the skew-symmetric matrix 119861
119879 Since
matrix 119861119879comes from triangulation of the Riemann surface
119878119892119899 119861119879ismutation finite see [3 p 18] the exchange relations
(3) take the following form
11990911199091015840
1= 1199092
2+ 1199092
3+ sdot sdot sdot + 119909
2
3119896
11990921199091015840
2= 1199092
1+ 1199092
3+ sdot sdot sdot + 119909
2
3119896
11990931198961199091015840
3119896= 1199092
1+ 1199092
2+ sdot sdot sdot + 119909
2
3119896minus1
(18)
One can construct the mutation tree T3119896
using relations(18) the reader is encouraged to verify that T
3119896is similar
middot middot middot middot middot middot middot middot middot middot middot middot middot middot middot middot middot middot
Figure 5 The Bratteli diagram of a cluster 119862lowast-algebra of rank 6
to the one shown in Figure 4 except for the number of theoutgoing edges at each vertex that is equal to 3119896
A tedious but straightforward calculation shows that theonly equivalent clusters at the same level of T
3119896are the ones
at the extremities of tuples (11990910158401 119909
1015840
3119896) in other words one
gets the following system of equivalences of clusters
12058313119896
(x) = 12058321(x)
12058323119896
(x) = 12058331(x)
1205833119896minus13119896
(x) = 12058331198961
(x)
(19)
where 120583119894119895(x) fl 120583
119895(120583119894(x))
The graph B(x 119861119879) is the quotient of T
3119896by the ℓ-
equivalence relations (19) for 119896 = 2 such a graph is sketchedin Figure 5 A(x 119876
119892119899) is an AF-algebra given by the Bratteli
diagramB(x 119861119879)
Lemma 15 The set
PrimA (x 119876119892119899) = 119868
120579| 120579 isin R
6119892minus7+2119899
is generic (20)
where A(x 119876119892119899)119868120579is an AF-algebra 119860
120579associated with the
convergent Jacobi-Perron continued fraction of vector (1 120579) seeSection 23
Proof We adapt the argument of [9 case 119896 = 1] to thecase 119896 ge 1 Let 119889 = 6119892 minus 6 + 2119899 be dimension of thespace 119879
119892119899 Roughly speaking the Bratteli diagram B(x 119861
119879)
of algebraA(x 119876119892119899) can be cut in two disjoint piecesG
120579and
B(x 119861119879)minusG120579 as it is shown by [9 Figure 7]G
120579is a (finite or
infinite) vertical strip of constant ldquowidthrdquo 119889 where 119889 is equalto the number of vertices cut from each level ofB(x 119861
119879)The
reader is encouraged to verify that G120579is exactly the Bratteli
diagram of the AF-algebra119860120579associated with the convergent
Jacobi-Perron continued fraction of a generic vector (1 120579) seeSection 23
On the other hand the complement B(x 119861119879) minus G
120579is
a hereditary Bratteli diagram which defines an ideal 119868120579of
algebra A(x 119876119892119899) such that
A (x 119876119892119899)
119868120579
= 119860120579 (21)
Journal of Function Spaces 7
see [6 Lemma 32] Moreover 119868120579is a primitive ideal [6
Theorem38] (An extra care is required if 120579 = (120579119894) is a rational
vector the complete argument can be found in [9 pp 980ndash985]) Lemma 15 follows
Lemma 16 The sequence of primitive ideals 119868120579119899
converges to119868120579in the Jacobson topology in PrimA(x 119876
119892119899) if and only if the
sequence 120579119899converges to 120579 in the Euclidean space R6119892minus7+2119899
Proof The proof is a straightforward adaption of the argu-ment in [9 pp 986ndash988] we leave it as an exercise to thereader
Let 120590119905 A(x 119876
119892119899) rarr A(x 119876
119892119899) be the Tomita-
Takesaki flow that is the group 120590119905| 119905 isin R of modular
automorphisms of algebra A(x 119876119892119899) see Section 4 Because
119868120579sub A(x 119876
119892119899) the image 120590119905(119868
120579) of 119868120579is correctly defined
for all 119905 isin R 120590119905(119868120579) is an ideal of A(x 119876
119892119899) but not
necessarily a primitive ideal Since 120590119905is an algebraic form
of the Teichmuller geodesic flow on the space 119879119892119899
[15] oneconcludes that the family of ideals
120590119905(119868120579) sub A (x 119876
119892119899) | 119905 isin R 120579 isin R
6119892minus7+2119899
(22)
can be taken for a coordinate system in the space 119879119892119899
cong
R6119892minus6+2119899 In view of Lemmas 15 and 16 one gets the requiredhomeomorphism
ℎ PrimA (x 119876119892119899) timesR
997888rarr 119880 sube 119879119892119899| 119880 is generic
(23)
such that the quotient algebra 119860120579= A(x 119876
119892119899)120590119905(119868120579) is a
noncommutative coordinate ring of theRiemann surface 119878119892119899
Theorem 2 is proved
4 An Analog of Modular Flow on A(x 119876119892119899)
41 Modular Automorphisms 120590119905| 119905 isin R Recall that the
Ptolemy relations (5) for the Penner coordinates 120582(120574119894) in
the space 119879119892119899
are homogeneous in particular the system119905120582(120574119894) | 119905 isin R of such coordinates will also satisfy the
Ptolemy relations On the other hand for the cluster 119862lowast-algebra A(x 119876
119892119899) the variables 119909
119894= 120582(120574
119894) and one gets an
obvious isomorphism A(x 119876119892119899) cong A(119905x 119876
119892119899) for all 119905 isin R
Since A(119905x 119876119892119899) sube A(x 119876
119892119899) one obtains a one-parameter
group of automorphisms
120590119905 A (x 119876
119892119899) 997888rarr A (x 119876
119892119899) (24)
By analogy with [21] we will call 120590119905a Tomita-Takesaki flow on
the cluster119862lowast-algebraA(x 119876119892119899)The reader is encouraged to
verify that120590119905is an algebraic formof the geodesic flow119879119905 on the
Teichmuller space 119879119892119899 see [15] for an introduction Roughly
speaking such a flow comes from the one-parameter groupof matrices
(
119890119905
0
0 119890minus119905
) (25)
acting on the space of holomorphic quadratic differentials ontheRiemann surface 119878
119892119899 the latter is known to be isomorphic
to the Teichmuller space 119879119892119899
42 Connes Invariant119879(A(x 119876119892119899)) Recall that an analogy of
theConnes invariant 119879(M) for a119862lowast-algebraM endowedwithamodular automorphism group120590
119905is the set119879(M) fl 119905 isin R |
120590119905is inner [21] The group of inner automorphisms of the
space 119879119892119899
and algebra A(x 119876119892119899) is isomorphic to the map-
ping class group mod 119878119892119899
of surface 119878119892119899 The automorphism
120601 isin mod119878119892119899
is called pseudo-Anosov if 120601(F120583) = 120582
120601F120583
where F120583is invariant measured foliation and 120582
120601gt 1 is
a constant called dilatation of 120601 120582120601is always an algebraic
number of the maximal degree 6119892 minus 6 + 2119899 [22] It is knownthat if 120601 isin mod119878
119892119899is pseudo-Anosov then there exists
a trajectory O of the geodesic flow 119879119905 and a point 119878
119892119899isin
119879119892119899 such that the points 119878
119892119899and 120601(119878
119892119899) belong to O [15]
O is called an axis of the pseudo-Anosov automorphism120601 The axis can be used to calculate the Connes invariant119879(A(x 119876
119892119899)) of the cluster 119862lowast-algebra A(x 119876
119892119899) indeed in
view of formula (25) one must solve the following system ofequations
120590119905(119909) = 119890
119905
119909
120601 (119909) = 120582120601119909
(26)
for a point 119909 isin O Thus 120590119905(119909) coincides with the inner
automorphism 120601(119909) if and only if 119905 = log 120582120601 Taking all
pseudo-Anosov automorphisms 120601 isin mod119878119892119899 one gets a
formula for the Connes invariant
119879 (A (x 119876119892119899))
= log 120582120601| 120601 isin mod 119878
119892119899is pseudo-Anosov
(27)
Remark 17 The Connes invariant (27) says that the family ofcluster 119862lowast-algebras A(x 119876
119892119899) is an analog of the type III
120582
factors of von Neumann algebras see [21]
Disclosure
All errors and misconceptions in this paper are solely theauthorrsquos
Competing Interests
The author declares that there are no competing interests inthe paper
Acknowledgments
It is the authorrsquos pleasure to thank Ibrahim Assem andthe SAG group of the Department of Mathematics of theUniversity of Sherbrooke for hospitality and excellent work-ing conditions The author is grateful to Ibrahim AssemThomas Brustle Daniele Mundici Ralf Schiffler and VasilisaShramchenko for an introduction to the wonderland ofcluster algebras and helpful correspondence
8 Journal of Function Spaces
References
[1] S Fomin and A Zelevinsky ldquoCluster algebras I foundationsrdquoJournal of the American Mathematical Society vol 15 no 2 pp497ndash529 2002
[2] S Fomin M Shapiro and D Thurston ldquoCluster algebras andtriangulated surfaces I Cluster complexesrdquo Acta Mathematicavol 201 no 1 pp 83ndash146 2008
[3] L K Williams ldquoCluster algebras an introductionrdquo Bulletin ofthe American Mathematical Society vol 51 no 1 pp 1ndash26 2014
[4] E G Effros Dimensions and Clowast-Algebras vol 46 of CBMSRegional Conference Series in Mathematics Conference Boardof the Mathematical Sciences 1981
[5] B Blackadar K-Theory for Operator Algebras vol 5 MSRIPublications Springer 1986
[6] O Bratteli ldquoInductive limits of finite dimensional Clowast-algebrasrdquoTransactions of the American Mathematical Society vol 171 pp195ndash234 1972
[7] R C Penner ldquoThe decorated Teichmuller space of puncturedsurfacesrdquoCommunications inMathematical Physics vol 113 no2 pp 299ndash339 1987
[8] D Mundici ldquoFarey stellar subdivisions ultrasimplicial groupsand119870
0of AF119862-algebrasrdquoAdvances inMathematics vol 68 no
1 pp 23ndash39 1988[9] F P Boca ldquoAnAF algebra associatedwith the Farey tessellationrdquo
Canadian Journal of Mathematics vol 60 no 5 pp 975ndash10002008
[10] D Mundici ldquoRevisiting the Farey AF-algebrardquoMilan Journal ofMathematics vol 79 no 2 pp 643ndash656 2011
[11] D Mundici ldquoRecognizing the Farey-Stern-Brocot AF algebraDedicated to the memory of Renato Caccioppolirdquo RendicontiLincei-Matematica e Applicazioni vol 20 pp 327ndash338 2009
[12] C Eckhardt ldquoA noncommutative Gauss maprdquo MathematicaScandinavica vol 108 no 2 pp 233ndash250 2011
[13] G Panti ldquoPrime ideals in free 119897 -groups and free vector latticesrdquoJournal of Algebra vol 219 no 1 pp 173ndash200 1999
[14] E G Effros and C L Shen ldquoApproximately finite 119862-algebrasand continued fractionsrdquo Indiana University Mathematics Jour-nal vol 29 no 2 pp 191ndash204 1980
[15] W A Veech ldquoThe Teichmuller geodesic flowrdquo Annals ofMathematics Second Series vol 124 no 3 pp 441ndash530 1986
[16] I VNikolaev ldquoOn aTeichmuller functor between the categoriesof complex tori and the Effros-Shen algebrasrdquoNew York Journalof Mathematics vol 15 pp 125ndash132 2009
[17] I Nikolaev ldquoRiemann surfaces and AF-algebrasrdquo Annals ofFunctional Analysis vol 7 no 2 pp 371ndash380 2016
[18] I Nikolaev ldquoCluster Clowast-algebras and knot polynomialsrdquo httparxivorgabs160301180
[19] I Nikolaev ldquoK-theory of cluster Clowast-algebrasrdquo httparxivorgabs151200276
[20] L Bernstein The Jacobi-Perron AlgorithmmdashIts Theory andApplication vol 207 of Lecture Notes in Mathematics SpringerBerlin Germany 1971
[21] A Connes ldquoVon Neumann algebrasrdquo in Proceedings of theInternational Congress of Mathematicians (ICM rsquo78) pp 97ndash109Helsinki Finland 1978
[22] W P Thurston ldquoOn the geometry and dynamics of diffeo-morphisms of surfacesrdquo Bulletin of the American MathematicalSociety vol 19 no 2 pp 417ndash431 1988
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Function Spaces 5
(x1 x21 x23+
x2 x3) (x1 x2 x21 x
22+
x3)( x22 x
23+
x1 x2 x3)
(x1 x2 x3)
(x1 x2 x3)(x1 x2 x3)
(x1 x2 x3)
(x1 x21 + ((x21 + x22) x3)2
x2x21 + x
22
x3)
( x22 + ((x21 + x22) x3)2
x1 x2
x21 + x
22
x3)( x
22 + x2
3
x1 x2((x22 + x
23) x1)2 + x
22
x3)
( x22 + x23
x1((x22 + x
23) x1)2 + x
23
x2 x3)
(((x21 + x23) x2)2 + x
23
x1x21 + x
23
x2 x3) (x1 x
21 + x
23
x2x21 + ((x21 + x
23) x2)2
x3)
Figure 4 The mutation tree
Let us verify that matrix 119861119879is mutation finite indeed for
each 119896 isin 1 2 3 the matrix mutation formula (4) gives us120583119896(119861119879) = minus119861
119879
Therefore the exchange relations (3) do not vary it isverified directly that such relations have the following form
11990911199091015840
1= 1199092
2+ 1199092
3
11990921199091015840
2= 1199092
1+ 1199092
3
11990931199091015840
3= 1199092
1+ 1199092
2
(10)
Consider a mutation tree T3shown in Figure 4 the
vertices of T3correspond to the mutations of cluster x =
(1199091 1199092 1199093) following the exchange rules (10)
The reader is encouraged to verify that modulo a cyclicpermutation of variables 1199091015840
1= 1199092 11990910158402= 1199093 11990910158403= 1199091and
1199091015840
1= 1199093 11990910158402= 1199091 11990910158403= 1199092one obtains (resp) the following
equivalences of clusters
12058313(x) = 120583
21(x)
12058323(x) = 120583
31(x)
(11)
where 120583119894119895(x) fl 120583
119895(120583119894(x)) there are no other cluster equiva-
lences for the vertices of the same level of graph T3
To determine the graph B(x 119861119879) one needs to take the
quotient of T3by the ℓ-equivalence relations (11) since the
pattern repeats for each level of T3 one gets the B(x 119861
119879)
shown in Figure 3 The cluster 119862lowast-algebra A(x 119861119879) is an AF-
algebra with the Bratteli diagramB(x 119861119879)
Notice that the Bratteli diagram B(x 119861119879) of our AF-
algebra A(x 119861119879) and such of the Mundici algebra M are
distinct compare [10 Figure 1] yet there is an obviousinclusion of one diagram into another Namely if one erasesa ldquocamelrsquos backrdquo (ie the two extreme sides of the diagram)in the Bratteli diagram of M then one gets exactly thediagram in Figure 3 Formally if G is the Bratteli diagramof the Mundici algebra M the complement G minus B(x 119861
119879)
is a hereditary Bratteli diagram which gives rise to an ideal1198680subM such that
A (x 119861119879) cong
M
1198680
(12)
see [6 Lemma 32] 1198680is a primitive ideal ibid Theorem 38
(It is interesting to calculate the group 1198700(1198680) in the context
of the work of [13])On the other hand the space PrimM (and hence
PrimA(x 119861119879)) is well understood see for example [13] or
[9 Proposition 7] Namely
Prim(
M
1198680
) = 119868120579| 120579 isin R (13)
where 119868120579sub M is such that M119868
120579cong 119860120579is the Effros-Shen
algebra [14] if 120579 is an irrational number or M119868120579cong 119872119902is
finite-dimensional matrix 119862lowast-algebra (and an extension ofsuch by the 119862lowast-algebra of compact operators) if 120579 = 119901119902 isa rational number (Note that the third series of primitiveideals of [9 Proposition 7] correspond to the ideal 119868
0)
6 Journal of Function Spaces
Moreover given the Jacobson topology on PrimM thereexists a homeomorphism
ℎ Prim(
M
1198680
) 997888rarr R (14)
defined by the formula 119868120579997891rarr 120579 see [9 Corollary 12]
Let 120590119905 M119868
0rarr M119868
0be the Tomita-Takesaki flow
that is a one-parameter automorphism group of M1198680 see
Section 4 Because 119868120579sub M119868
0 the image 120590119905(119868
120579) of 119868
120579is
correctly defined for all 119905 isin R 120590119905(119868120579) is an ideal of M119868
0
but not necessarily primitive Since 120590119905is nothing but (an
algebraic form of) the Teichmuller geodesic flow on 11987911
[15]one concludes that that the family of ideals
120590119905(119868120579) sub
M
1198680
| 119905 isin R 120579 isin R (15)
can be taken for a coordinate system in the space 11987911cong R2
In view of (14) andM1198680cong A(x 119876
11) one gets the required
homeomorphism
ℎ PrimA (x 11987611) timesR 997888rarr 119879
11 (16)
such that the quotient algebra A(x 11987611)120590119905(119868120579) is a noncom-
mutative coordinate ring of the Riemann surface 11987811
Remark 14 The family of algebras A(x 11987611)120590119905(119868120579) | 120579 =
Const 119905 isin R are in general pairwise nonisomorphic (Forotherwise all ideals 120590
119905(119868120579) | 119905 isin R were primitive) However
theGrothendieck semigroups119870+0are isomorphic see [14] the
action of 120590119905is given by the following formula (see Section 4)
119870+
0(
A (x 11987611)
120590119905(119868120579)
) cong 119890119905
(Z + Z120579) (17)
(ii) The general case 119898 = 3119896 = 3(2119892 minus 2 + 119899) is treatedlikewise Notice that if 119889 = 6119892 minus 6 + 2119899 is dimension of thespace119879
119892119899 thenwe have119898minus119889 = 119899 in particular rank119898 of the
cluster 119862lowast-algebra A(x 119876119892119899) determines completely the pair
(119892 119899) provided 119889 is a fixed constant (If 119889 is not fixed thereis only a finite number of different pairs (119892 119899) for given rank119898)
Let (x 119861119879) be the seed given by the cluster x =
(1199091 119909
3119896) and the skew-symmetric matrix 119861
119879 Since
matrix 119861119879comes from triangulation of the Riemann surface
119878119892119899 119861119879ismutation finite see [3 p 18] the exchange relations
(3) take the following form
11990911199091015840
1= 1199092
2+ 1199092
3+ sdot sdot sdot + 119909
2
3119896
11990921199091015840
2= 1199092
1+ 1199092
3+ sdot sdot sdot + 119909
2
3119896
11990931198961199091015840
3119896= 1199092
1+ 1199092
2+ sdot sdot sdot + 119909
2
3119896minus1
(18)
One can construct the mutation tree T3119896
using relations(18) the reader is encouraged to verify that T
3119896is similar
middot middot middot middot middot middot middot middot middot middot middot middot middot middot middot middot middot middot
Figure 5 The Bratteli diagram of a cluster 119862lowast-algebra of rank 6
to the one shown in Figure 4 except for the number of theoutgoing edges at each vertex that is equal to 3119896
A tedious but straightforward calculation shows that theonly equivalent clusters at the same level of T
3119896are the ones
at the extremities of tuples (11990910158401 119909
1015840
3119896) in other words one
gets the following system of equivalences of clusters
12058313119896
(x) = 12058321(x)
12058323119896
(x) = 12058331(x)
1205833119896minus13119896
(x) = 12058331198961
(x)
(19)
where 120583119894119895(x) fl 120583
119895(120583119894(x))
The graph B(x 119861119879) is the quotient of T
3119896by the ℓ-
equivalence relations (19) for 119896 = 2 such a graph is sketchedin Figure 5 A(x 119876
119892119899) is an AF-algebra given by the Bratteli
diagramB(x 119861119879)
Lemma 15 The set
PrimA (x 119876119892119899) = 119868
120579| 120579 isin R
6119892minus7+2119899
is generic (20)
where A(x 119876119892119899)119868120579is an AF-algebra 119860
120579associated with the
convergent Jacobi-Perron continued fraction of vector (1 120579) seeSection 23
Proof We adapt the argument of [9 case 119896 = 1] to thecase 119896 ge 1 Let 119889 = 6119892 minus 6 + 2119899 be dimension of thespace 119879
119892119899 Roughly speaking the Bratteli diagram B(x 119861
119879)
of algebraA(x 119876119892119899) can be cut in two disjoint piecesG
120579and
B(x 119861119879)minusG120579 as it is shown by [9 Figure 7]G
120579is a (finite or
infinite) vertical strip of constant ldquowidthrdquo 119889 where 119889 is equalto the number of vertices cut from each level ofB(x 119861
119879)The
reader is encouraged to verify that G120579is exactly the Bratteli
diagram of the AF-algebra119860120579associated with the convergent
Jacobi-Perron continued fraction of a generic vector (1 120579) seeSection 23
On the other hand the complement B(x 119861119879) minus G
120579is
a hereditary Bratteli diagram which defines an ideal 119868120579of
algebra A(x 119876119892119899) such that
A (x 119876119892119899)
119868120579
= 119860120579 (21)
Journal of Function Spaces 7
see [6 Lemma 32] Moreover 119868120579is a primitive ideal [6
Theorem38] (An extra care is required if 120579 = (120579119894) is a rational
vector the complete argument can be found in [9 pp 980ndash985]) Lemma 15 follows
Lemma 16 The sequence of primitive ideals 119868120579119899
converges to119868120579in the Jacobson topology in PrimA(x 119876
119892119899) if and only if the
sequence 120579119899converges to 120579 in the Euclidean space R6119892minus7+2119899
Proof The proof is a straightforward adaption of the argu-ment in [9 pp 986ndash988] we leave it as an exercise to thereader
Let 120590119905 A(x 119876
119892119899) rarr A(x 119876
119892119899) be the Tomita-
Takesaki flow that is the group 120590119905| 119905 isin R of modular
automorphisms of algebra A(x 119876119892119899) see Section 4 Because
119868120579sub A(x 119876
119892119899) the image 120590119905(119868
120579) of 119868120579is correctly defined
for all 119905 isin R 120590119905(119868120579) is an ideal of A(x 119876
119892119899) but not
necessarily a primitive ideal Since 120590119905is an algebraic form
of the Teichmuller geodesic flow on the space 119879119892119899
[15] oneconcludes that the family of ideals
120590119905(119868120579) sub A (x 119876
119892119899) | 119905 isin R 120579 isin R
6119892minus7+2119899
(22)
can be taken for a coordinate system in the space 119879119892119899
cong
R6119892minus6+2119899 In view of Lemmas 15 and 16 one gets the requiredhomeomorphism
ℎ PrimA (x 119876119892119899) timesR
997888rarr 119880 sube 119879119892119899| 119880 is generic
(23)
such that the quotient algebra 119860120579= A(x 119876
119892119899)120590119905(119868120579) is a
noncommutative coordinate ring of theRiemann surface 119878119892119899
Theorem 2 is proved
4 An Analog of Modular Flow on A(x 119876119892119899)
41 Modular Automorphisms 120590119905| 119905 isin R Recall that the
Ptolemy relations (5) for the Penner coordinates 120582(120574119894) in
the space 119879119892119899
are homogeneous in particular the system119905120582(120574119894) | 119905 isin R of such coordinates will also satisfy the
Ptolemy relations On the other hand for the cluster 119862lowast-algebra A(x 119876
119892119899) the variables 119909
119894= 120582(120574
119894) and one gets an
obvious isomorphism A(x 119876119892119899) cong A(119905x 119876
119892119899) for all 119905 isin R
Since A(119905x 119876119892119899) sube A(x 119876
119892119899) one obtains a one-parameter
group of automorphisms
120590119905 A (x 119876
119892119899) 997888rarr A (x 119876
119892119899) (24)
By analogy with [21] we will call 120590119905a Tomita-Takesaki flow on
the cluster119862lowast-algebraA(x 119876119892119899)The reader is encouraged to
verify that120590119905is an algebraic formof the geodesic flow119879119905 on the
Teichmuller space 119879119892119899 see [15] for an introduction Roughly
speaking such a flow comes from the one-parameter groupof matrices
(
119890119905
0
0 119890minus119905
) (25)
acting on the space of holomorphic quadratic differentials ontheRiemann surface 119878
119892119899 the latter is known to be isomorphic
to the Teichmuller space 119879119892119899
42 Connes Invariant119879(A(x 119876119892119899)) Recall that an analogy of
theConnes invariant 119879(M) for a119862lowast-algebraM endowedwithamodular automorphism group120590
119905is the set119879(M) fl 119905 isin R |
120590119905is inner [21] The group of inner automorphisms of the
space 119879119892119899
and algebra A(x 119876119892119899) is isomorphic to the map-
ping class group mod 119878119892119899
of surface 119878119892119899 The automorphism
120601 isin mod119878119892119899
is called pseudo-Anosov if 120601(F120583) = 120582
120601F120583
where F120583is invariant measured foliation and 120582
120601gt 1 is
a constant called dilatation of 120601 120582120601is always an algebraic
number of the maximal degree 6119892 minus 6 + 2119899 [22] It is knownthat if 120601 isin mod119878
119892119899is pseudo-Anosov then there exists
a trajectory O of the geodesic flow 119879119905 and a point 119878
119892119899isin
119879119892119899 such that the points 119878
119892119899and 120601(119878
119892119899) belong to O [15]
O is called an axis of the pseudo-Anosov automorphism120601 The axis can be used to calculate the Connes invariant119879(A(x 119876
119892119899)) of the cluster 119862lowast-algebra A(x 119876
119892119899) indeed in
view of formula (25) one must solve the following system ofequations
120590119905(119909) = 119890
119905
119909
120601 (119909) = 120582120601119909
(26)
for a point 119909 isin O Thus 120590119905(119909) coincides with the inner
automorphism 120601(119909) if and only if 119905 = log 120582120601 Taking all
pseudo-Anosov automorphisms 120601 isin mod119878119892119899 one gets a
formula for the Connes invariant
119879 (A (x 119876119892119899))
= log 120582120601| 120601 isin mod 119878
119892119899is pseudo-Anosov
(27)
Remark 17 The Connes invariant (27) says that the family ofcluster 119862lowast-algebras A(x 119876
119892119899) is an analog of the type III
120582
factors of von Neumann algebras see [21]
Disclosure
All errors and misconceptions in this paper are solely theauthorrsquos
Competing Interests
The author declares that there are no competing interests inthe paper
Acknowledgments
It is the authorrsquos pleasure to thank Ibrahim Assem andthe SAG group of the Department of Mathematics of theUniversity of Sherbrooke for hospitality and excellent work-ing conditions The author is grateful to Ibrahim AssemThomas Brustle Daniele Mundici Ralf Schiffler and VasilisaShramchenko for an introduction to the wonderland ofcluster algebras and helpful correspondence
8 Journal of Function Spaces
References
[1] S Fomin and A Zelevinsky ldquoCluster algebras I foundationsrdquoJournal of the American Mathematical Society vol 15 no 2 pp497ndash529 2002
[2] S Fomin M Shapiro and D Thurston ldquoCluster algebras andtriangulated surfaces I Cluster complexesrdquo Acta Mathematicavol 201 no 1 pp 83ndash146 2008
[3] L K Williams ldquoCluster algebras an introductionrdquo Bulletin ofthe American Mathematical Society vol 51 no 1 pp 1ndash26 2014
[4] E G Effros Dimensions and Clowast-Algebras vol 46 of CBMSRegional Conference Series in Mathematics Conference Boardof the Mathematical Sciences 1981
[5] B Blackadar K-Theory for Operator Algebras vol 5 MSRIPublications Springer 1986
[6] O Bratteli ldquoInductive limits of finite dimensional Clowast-algebrasrdquoTransactions of the American Mathematical Society vol 171 pp195ndash234 1972
[7] R C Penner ldquoThe decorated Teichmuller space of puncturedsurfacesrdquoCommunications inMathematical Physics vol 113 no2 pp 299ndash339 1987
[8] D Mundici ldquoFarey stellar subdivisions ultrasimplicial groupsand119870
0of AF119862-algebrasrdquoAdvances inMathematics vol 68 no
1 pp 23ndash39 1988[9] F P Boca ldquoAnAF algebra associatedwith the Farey tessellationrdquo
Canadian Journal of Mathematics vol 60 no 5 pp 975ndash10002008
[10] D Mundici ldquoRevisiting the Farey AF-algebrardquoMilan Journal ofMathematics vol 79 no 2 pp 643ndash656 2011
[11] D Mundici ldquoRecognizing the Farey-Stern-Brocot AF algebraDedicated to the memory of Renato Caccioppolirdquo RendicontiLincei-Matematica e Applicazioni vol 20 pp 327ndash338 2009
[12] C Eckhardt ldquoA noncommutative Gauss maprdquo MathematicaScandinavica vol 108 no 2 pp 233ndash250 2011
[13] G Panti ldquoPrime ideals in free 119897 -groups and free vector latticesrdquoJournal of Algebra vol 219 no 1 pp 173ndash200 1999
[14] E G Effros and C L Shen ldquoApproximately finite 119862-algebrasand continued fractionsrdquo Indiana University Mathematics Jour-nal vol 29 no 2 pp 191ndash204 1980
[15] W A Veech ldquoThe Teichmuller geodesic flowrdquo Annals ofMathematics Second Series vol 124 no 3 pp 441ndash530 1986
[16] I VNikolaev ldquoOn aTeichmuller functor between the categoriesof complex tori and the Effros-Shen algebrasrdquoNew York Journalof Mathematics vol 15 pp 125ndash132 2009
[17] I Nikolaev ldquoRiemann surfaces and AF-algebrasrdquo Annals ofFunctional Analysis vol 7 no 2 pp 371ndash380 2016
[18] I Nikolaev ldquoCluster Clowast-algebras and knot polynomialsrdquo httparxivorgabs160301180
[19] I Nikolaev ldquoK-theory of cluster Clowast-algebrasrdquo httparxivorgabs151200276
[20] L Bernstein The Jacobi-Perron AlgorithmmdashIts Theory andApplication vol 207 of Lecture Notes in Mathematics SpringerBerlin Germany 1971
[21] A Connes ldquoVon Neumann algebrasrdquo in Proceedings of theInternational Congress of Mathematicians (ICM rsquo78) pp 97ndash109Helsinki Finland 1978
[22] W P Thurston ldquoOn the geometry and dynamics of diffeo-morphisms of surfacesrdquo Bulletin of the American MathematicalSociety vol 19 no 2 pp 417ndash431 1988
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Journal of Function Spaces
Moreover given the Jacobson topology on PrimM thereexists a homeomorphism
ℎ Prim(
M
1198680
) 997888rarr R (14)
defined by the formula 119868120579997891rarr 120579 see [9 Corollary 12]
Let 120590119905 M119868
0rarr M119868
0be the Tomita-Takesaki flow
that is a one-parameter automorphism group of M1198680 see
Section 4 Because 119868120579sub M119868
0 the image 120590119905(119868
120579) of 119868
120579is
correctly defined for all 119905 isin R 120590119905(119868120579) is an ideal of M119868
0
but not necessarily primitive Since 120590119905is nothing but (an
algebraic form of) the Teichmuller geodesic flow on 11987911
[15]one concludes that that the family of ideals
120590119905(119868120579) sub
M
1198680
| 119905 isin R 120579 isin R (15)
can be taken for a coordinate system in the space 11987911cong R2
In view of (14) andM1198680cong A(x 119876
11) one gets the required
homeomorphism
ℎ PrimA (x 11987611) timesR 997888rarr 119879
11 (16)
such that the quotient algebra A(x 11987611)120590119905(119868120579) is a noncom-
mutative coordinate ring of the Riemann surface 11987811
Remark 14 The family of algebras A(x 11987611)120590119905(119868120579) | 120579 =
Const 119905 isin R are in general pairwise nonisomorphic (Forotherwise all ideals 120590
119905(119868120579) | 119905 isin R were primitive) However
theGrothendieck semigroups119870+0are isomorphic see [14] the
action of 120590119905is given by the following formula (see Section 4)
119870+
0(
A (x 11987611)
120590119905(119868120579)
) cong 119890119905
(Z + Z120579) (17)
(ii) The general case 119898 = 3119896 = 3(2119892 minus 2 + 119899) is treatedlikewise Notice that if 119889 = 6119892 minus 6 + 2119899 is dimension of thespace119879
119892119899 thenwe have119898minus119889 = 119899 in particular rank119898 of the
cluster 119862lowast-algebra A(x 119876119892119899) determines completely the pair
(119892 119899) provided 119889 is a fixed constant (If 119889 is not fixed thereis only a finite number of different pairs (119892 119899) for given rank119898)
Let (x 119861119879) be the seed given by the cluster x =
(1199091 119909
3119896) and the skew-symmetric matrix 119861
119879 Since
matrix 119861119879comes from triangulation of the Riemann surface
119878119892119899 119861119879ismutation finite see [3 p 18] the exchange relations
(3) take the following form
11990911199091015840
1= 1199092
2+ 1199092
3+ sdot sdot sdot + 119909
2
3119896
11990921199091015840
2= 1199092
1+ 1199092
3+ sdot sdot sdot + 119909
2
3119896
11990931198961199091015840
3119896= 1199092
1+ 1199092
2+ sdot sdot sdot + 119909
2
3119896minus1
(18)
One can construct the mutation tree T3119896
using relations(18) the reader is encouraged to verify that T
3119896is similar
middot middot middot middot middot middot middot middot middot middot middot middot middot middot middot middot middot middot
Figure 5 The Bratteli diagram of a cluster 119862lowast-algebra of rank 6
to the one shown in Figure 4 except for the number of theoutgoing edges at each vertex that is equal to 3119896
A tedious but straightforward calculation shows that theonly equivalent clusters at the same level of T
3119896are the ones
at the extremities of tuples (11990910158401 119909
1015840
3119896) in other words one
gets the following system of equivalences of clusters
12058313119896
(x) = 12058321(x)
12058323119896
(x) = 12058331(x)
1205833119896minus13119896
(x) = 12058331198961
(x)
(19)
where 120583119894119895(x) fl 120583
119895(120583119894(x))
The graph B(x 119861119879) is the quotient of T
3119896by the ℓ-
equivalence relations (19) for 119896 = 2 such a graph is sketchedin Figure 5 A(x 119876
119892119899) is an AF-algebra given by the Bratteli
diagramB(x 119861119879)
Lemma 15 The set
PrimA (x 119876119892119899) = 119868
120579| 120579 isin R
6119892minus7+2119899
is generic (20)
where A(x 119876119892119899)119868120579is an AF-algebra 119860
120579associated with the
convergent Jacobi-Perron continued fraction of vector (1 120579) seeSection 23
Proof We adapt the argument of [9 case 119896 = 1] to thecase 119896 ge 1 Let 119889 = 6119892 minus 6 + 2119899 be dimension of thespace 119879
119892119899 Roughly speaking the Bratteli diagram B(x 119861
119879)
of algebraA(x 119876119892119899) can be cut in two disjoint piecesG
120579and
B(x 119861119879)minusG120579 as it is shown by [9 Figure 7]G
120579is a (finite or
infinite) vertical strip of constant ldquowidthrdquo 119889 where 119889 is equalto the number of vertices cut from each level ofB(x 119861
119879)The
reader is encouraged to verify that G120579is exactly the Bratteli
diagram of the AF-algebra119860120579associated with the convergent
Jacobi-Perron continued fraction of a generic vector (1 120579) seeSection 23
On the other hand the complement B(x 119861119879) minus G
120579is
a hereditary Bratteli diagram which defines an ideal 119868120579of
algebra A(x 119876119892119899) such that
A (x 119876119892119899)
119868120579
= 119860120579 (21)
Journal of Function Spaces 7
see [6 Lemma 32] Moreover 119868120579is a primitive ideal [6
Theorem38] (An extra care is required if 120579 = (120579119894) is a rational
vector the complete argument can be found in [9 pp 980ndash985]) Lemma 15 follows
Lemma 16 The sequence of primitive ideals 119868120579119899
converges to119868120579in the Jacobson topology in PrimA(x 119876
119892119899) if and only if the
sequence 120579119899converges to 120579 in the Euclidean space R6119892minus7+2119899
Proof The proof is a straightforward adaption of the argu-ment in [9 pp 986ndash988] we leave it as an exercise to thereader
Let 120590119905 A(x 119876
119892119899) rarr A(x 119876
119892119899) be the Tomita-
Takesaki flow that is the group 120590119905| 119905 isin R of modular
automorphisms of algebra A(x 119876119892119899) see Section 4 Because
119868120579sub A(x 119876
119892119899) the image 120590119905(119868
120579) of 119868120579is correctly defined
for all 119905 isin R 120590119905(119868120579) is an ideal of A(x 119876
119892119899) but not
necessarily a primitive ideal Since 120590119905is an algebraic form
of the Teichmuller geodesic flow on the space 119879119892119899
[15] oneconcludes that the family of ideals
120590119905(119868120579) sub A (x 119876
119892119899) | 119905 isin R 120579 isin R
6119892minus7+2119899
(22)
can be taken for a coordinate system in the space 119879119892119899
cong
R6119892minus6+2119899 In view of Lemmas 15 and 16 one gets the requiredhomeomorphism
ℎ PrimA (x 119876119892119899) timesR
997888rarr 119880 sube 119879119892119899| 119880 is generic
(23)
such that the quotient algebra 119860120579= A(x 119876
119892119899)120590119905(119868120579) is a
noncommutative coordinate ring of theRiemann surface 119878119892119899
Theorem 2 is proved
4 An Analog of Modular Flow on A(x 119876119892119899)
41 Modular Automorphisms 120590119905| 119905 isin R Recall that the
Ptolemy relations (5) for the Penner coordinates 120582(120574119894) in
the space 119879119892119899
are homogeneous in particular the system119905120582(120574119894) | 119905 isin R of such coordinates will also satisfy the
Ptolemy relations On the other hand for the cluster 119862lowast-algebra A(x 119876
119892119899) the variables 119909
119894= 120582(120574
119894) and one gets an
obvious isomorphism A(x 119876119892119899) cong A(119905x 119876
119892119899) for all 119905 isin R
Since A(119905x 119876119892119899) sube A(x 119876
119892119899) one obtains a one-parameter
group of automorphisms
120590119905 A (x 119876
119892119899) 997888rarr A (x 119876
119892119899) (24)
By analogy with [21] we will call 120590119905a Tomita-Takesaki flow on
the cluster119862lowast-algebraA(x 119876119892119899)The reader is encouraged to
verify that120590119905is an algebraic formof the geodesic flow119879119905 on the
Teichmuller space 119879119892119899 see [15] for an introduction Roughly
speaking such a flow comes from the one-parameter groupof matrices
(
119890119905
0
0 119890minus119905
) (25)
acting on the space of holomorphic quadratic differentials ontheRiemann surface 119878
119892119899 the latter is known to be isomorphic
to the Teichmuller space 119879119892119899
42 Connes Invariant119879(A(x 119876119892119899)) Recall that an analogy of
theConnes invariant 119879(M) for a119862lowast-algebraM endowedwithamodular automorphism group120590
119905is the set119879(M) fl 119905 isin R |
120590119905is inner [21] The group of inner automorphisms of the
space 119879119892119899
and algebra A(x 119876119892119899) is isomorphic to the map-
ping class group mod 119878119892119899
of surface 119878119892119899 The automorphism
120601 isin mod119878119892119899
is called pseudo-Anosov if 120601(F120583) = 120582
120601F120583
where F120583is invariant measured foliation and 120582
120601gt 1 is
a constant called dilatation of 120601 120582120601is always an algebraic
number of the maximal degree 6119892 minus 6 + 2119899 [22] It is knownthat if 120601 isin mod119878
119892119899is pseudo-Anosov then there exists
a trajectory O of the geodesic flow 119879119905 and a point 119878
119892119899isin
119879119892119899 such that the points 119878
119892119899and 120601(119878
119892119899) belong to O [15]
O is called an axis of the pseudo-Anosov automorphism120601 The axis can be used to calculate the Connes invariant119879(A(x 119876
119892119899)) of the cluster 119862lowast-algebra A(x 119876
119892119899) indeed in
view of formula (25) one must solve the following system ofequations
120590119905(119909) = 119890
119905
119909
120601 (119909) = 120582120601119909
(26)
for a point 119909 isin O Thus 120590119905(119909) coincides with the inner
automorphism 120601(119909) if and only if 119905 = log 120582120601 Taking all
pseudo-Anosov automorphisms 120601 isin mod119878119892119899 one gets a
formula for the Connes invariant
119879 (A (x 119876119892119899))
= log 120582120601| 120601 isin mod 119878
119892119899is pseudo-Anosov
(27)
Remark 17 The Connes invariant (27) says that the family ofcluster 119862lowast-algebras A(x 119876
119892119899) is an analog of the type III
120582
factors of von Neumann algebras see [21]
Disclosure
All errors and misconceptions in this paper are solely theauthorrsquos
Competing Interests
The author declares that there are no competing interests inthe paper
Acknowledgments
It is the authorrsquos pleasure to thank Ibrahim Assem andthe SAG group of the Department of Mathematics of theUniversity of Sherbrooke for hospitality and excellent work-ing conditions The author is grateful to Ibrahim AssemThomas Brustle Daniele Mundici Ralf Schiffler and VasilisaShramchenko for an introduction to the wonderland ofcluster algebras and helpful correspondence
8 Journal of Function Spaces
References
[1] S Fomin and A Zelevinsky ldquoCluster algebras I foundationsrdquoJournal of the American Mathematical Society vol 15 no 2 pp497ndash529 2002
[2] S Fomin M Shapiro and D Thurston ldquoCluster algebras andtriangulated surfaces I Cluster complexesrdquo Acta Mathematicavol 201 no 1 pp 83ndash146 2008
[3] L K Williams ldquoCluster algebras an introductionrdquo Bulletin ofthe American Mathematical Society vol 51 no 1 pp 1ndash26 2014
[4] E G Effros Dimensions and Clowast-Algebras vol 46 of CBMSRegional Conference Series in Mathematics Conference Boardof the Mathematical Sciences 1981
[5] B Blackadar K-Theory for Operator Algebras vol 5 MSRIPublications Springer 1986
[6] O Bratteli ldquoInductive limits of finite dimensional Clowast-algebrasrdquoTransactions of the American Mathematical Society vol 171 pp195ndash234 1972
[7] R C Penner ldquoThe decorated Teichmuller space of puncturedsurfacesrdquoCommunications inMathematical Physics vol 113 no2 pp 299ndash339 1987
[8] D Mundici ldquoFarey stellar subdivisions ultrasimplicial groupsand119870
0of AF119862-algebrasrdquoAdvances inMathematics vol 68 no
1 pp 23ndash39 1988[9] F P Boca ldquoAnAF algebra associatedwith the Farey tessellationrdquo
Canadian Journal of Mathematics vol 60 no 5 pp 975ndash10002008
[10] D Mundici ldquoRevisiting the Farey AF-algebrardquoMilan Journal ofMathematics vol 79 no 2 pp 643ndash656 2011
[11] D Mundici ldquoRecognizing the Farey-Stern-Brocot AF algebraDedicated to the memory of Renato Caccioppolirdquo RendicontiLincei-Matematica e Applicazioni vol 20 pp 327ndash338 2009
[12] C Eckhardt ldquoA noncommutative Gauss maprdquo MathematicaScandinavica vol 108 no 2 pp 233ndash250 2011
[13] G Panti ldquoPrime ideals in free 119897 -groups and free vector latticesrdquoJournal of Algebra vol 219 no 1 pp 173ndash200 1999
[14] E G Effros and C L Shen ldquoApproximately finite 119862-algebrasand continued fractionsrdquo Indiana University Mathematics Jour-nal vol 29 no 2 pp 191ndash204 1980
[15] W A Veech ldquoThe Teichmuller geodesic flowrdquo Annals ofMathematics Second Series vol 124 no 3 pp 441ndash530 1986
[16] I VNikolaev ldquoOn aTeichmuller functor between the categoriesof complex tori and the Effros-Shen algebrasrdquoNew York Journalof Mathematics vol 15 pp 125ndash132 2009
[17] I Nikolaev ldquoRiemann surfaces and AF-algebrasrdquo Annals ofFunctional Analysis vol 7 no 2 pp 371ndash380 2016
[18] I Nikolaev ldquoCluster Clowast-algebras and knot polynomialsrdquo httparxivorgabs160301180
[19] I Nikolaev ldquoK-theory of cluster Clowast-algebrasrdquo httparxivorgabs151200276
[20] L Bernstein The Jacobi-Perron AlgorithmmdashIts Theory andApplication vol 207 of Lecture Notes in Mathematics SpringerBerlin Germany 1971
[21] A Connes ldquoVon Neumann algebrasrdquo in Proceedings of theInternational Congress of Mathematicians (ICM rsquo78) pp 97ndash109Helsinki Finland 1978
[22] W P Thurston ldquoOn the geometry and dynamics of diffeo-morphisms of surfacesrdquo Bulletin of the American MathematicalSociety vol 19 no 2 pp 417ndash431 1988
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Function Spaces 7
see [6 Lemma 32] Moreover 119868120579is a primitive ideal [6
Theorem38] (An extra care is required if 120579 = (120579119894) is a rational
vector the complete argument can be found in [9 pp 980ndash985]) Lemma 15 follows
Lemma 16 The sequence of primitive ideals 119868120579119899
converges to119868120579in the Jacobson topology in PrimA(x 119876
119892119899) if and only if the
sequence 120579119899converges to 120579 in the Euclidean space R6119892minus7+2119899
Proof The proof is a straightforward adaption of the argu-ment in [9 pp 986ndash988] we leave it as an exercise to thereader
Let 120590119905 A(x 119876
119892119899) rarr A(x 119876
119892119899) be the Tomita-
Takesaki flow that is the group 120590119905| 119905 isin R of modular
automorphisms of algebra A(x 119876119892119899) see Section 4 Because
119868120579sub A(x 119876
119892119899) the image 120590119905(119868
120579) of 119868120579is correctly defined
for all 119905 isin R 120590119905(119868120579) is an ideal of A(x 119876
119892119899) but not
necessarily a primitive ideal Since 120590119905is an algebraic form
of the Teichmuller geodesic flow on the space 119879119892119899
[15] oneconcludes that the family of ideals
120590119905(119868120579) sub A (x 119876
119892119899) | 119905 isin R 120579 isin R
6119892minus7+2119899
(22)
can be taken for a coordinate system in the space 119879119892119899
cong
R6119892minus6+2119899 In view of Lemmas 15 and 16 one gets the requiredhomeomorphism
ℎ PrimA (x 119876119892119899) timesR
997888rarr 119880 sube 119879119892119899| 119880 is generic
(23)
such that the quotient algebra 119860120579= A(x 119876
119892119899)120590119905(119868120579) is a
noncommutative coordinate ring of theRiemann surface 119878119892119899
Theorem 2 is proved
4 An Analog of Modular Flow on A(x 119876119892119899)
41 Modular Automorphisms 120590119905| 119905 isin R Recall that the
Ptolemy relations (5) for the Penner coordinates 120582(120574119894) in
the space 119879119892119899
are homogeneous in particular the system119905120582(120574119894) | 119905 isin R of such coordinates will also satisfy the
Ptolemy relations On the other hand for the cluster 119862lowast-algebra A(x 119876
119892119899) the variables 119909
119894= 120582(120574
119894) and one gets an
obvious isomorphism A(x 119876119892119899) cong A(119905x 119876
119892119899) for all 119905 isin R
Since A(119905x 119876119892119899) sube A(x 119876
119892119899) one obtains a one-parameter
group of automorphisms
120590119905 A (x 119876
119892119899) 997888rarr A (x 119876
119892119899) (24)
By analogy with [21] we will call 120590119905a Tomita-Takesaki flow on
the cluster119862lowast-algebraA(x 119876119892119899)The reader is encouraged to
verify that120590119905is an algebraic formof the geodesic flow119879119905 on the
Teichmuller space 119879119892119899 see [15] for an introduction Roughly
speaking such a flow comes from the one-parameter groupof matrices
(
119890119905
0
0 119890minus119905
) (25)
acting on the space of holomorphic quadratic differentials ontheRiemann surface 119878
119892119899 the latter is known to be isomorphic
to the Teichmuller space 119879119892119899
42 Connes Invariant119879(A(x 119876119892119899)) Recall that an analogy of
theConnes invariant 119879(M) for a119862lowast-algebraM endowedwithamodular automorphism group120590
119905is the set119879(M) fl 119905 isin R |
120590119905is inner [21] The group of inner automorphisms of the
space 119879119892119899
and algebra A(x 119876119892119899) is isomorphic to the map-
ping class group mod 119878119892119899
of surface 119878119892119899 The automorphism
120601 isin mod119878119892119899
is called pseudo-Anosov if 120601(F120583) = 120582
120601F120583
where F120583is invariant measured foliation and 120582
120601gt 1 is
a constant called dilatation of 120601 120582120601is always an algebraic
number of the maximal degree 6119892 minus 6 + 2119899 [22] It is knownthat if 120601 isin mod119878
119892119899is pseudo-Anosov then there exists
a trajectory O of the geodesic flow 119879119905 and a point 119878
119892119899isin
119879119892119899 such that the points 119878
119892119899and 120601(119878
119892119899) belong to O [15]
O is called an axis of the pseudo-Anosov automorphism120601 The axis can be used to calculate the Connes invariant119879(A(x 119876
119892119899)) of the cluster 119862lowast-algebra A(x 119876
119892119899) indeed in
view of formula (25) one must solve the following system ofequations
120590119905(119909) = 119890
119905
119909
120601 (119909) = 120582120601119909
(26)
for a point 119909 isin O Thus 120590119905(119909) coincides with the inner
automorphism 120601(119909) if and only if 119905 = log 120582120601 Taking all
pseudo-Anosov automorphisms 120601 isin mod119878119892119899 one gets a
formula for the Connes invariant
119879 (A (x 119876119892119899))
= log 120582120601| 120601 isin mod 119878
119892119899is pseudo-Anosov
(27)
Remark 17 The Connes invariant (27) says that the family ofcluster 119862lowast-algebras A(x 119876
119892119899) is an analog of the type III
120582
factors of von Neumann algebras see [21]
Disclosure
All errors and misconceptions in this paper are solely theauthorrsquos
Competing Interests
The author declares that there are no competing interests inthe paper
Acknowledgments
It is the authorrsquos pleasure to thank Ibrahim Assem andthe SAG group of the Department of Mathematics of theUniversity of Sherbrooke for hospitality and excellent work-ing conditions The author is grateful to Ibrahim AssemThomas Brustle Daniele Mundici Ralf Schiffler and VasilisaShramchenko for an introduction to the wonderland ofcluster algebras and helpful correspondence
8 Journal of Function Spaces
References
[1] S Fomin and A Zelevinsky ldquoCluster algebras I foundationsrdquoJournal of the American Mathematical Society vol 15 no 2 pp497ndash529 2002
[2] S Fomin M Shapiro and D Thurston ldquoCluster algebras andtriangulated surfaces I Cluster complexesrdquo Acta Mathematicavol 201 no 1 pp 83ndash146 2008
[3] L K Williams ldquoCluster algebras an introductionrdquo Bulletin ofthe American Mathematical Society vol 51 no 1 pp 1ndash26 2014
[4] E G Effros Dimensions and Clowast-Algebras vol 46 of CBMSRegional Conference Series in Mathematics Conference Boardof the Mathematical Sciences 1981
[5] B Blackadar K-Theory for Operator Algebras vol 5 MSRIPublications Springer 1986
[6] O Bratteli ldquoInductive limits of finite dimensional Clowast-algebrasrdquoTransactions of the American Mathematical Society vol 171 pp195ndash234 1972
[7] R C Penner ldquoThe decorated Teichmuller space of puncturedsurfacesrdquoCommunications inMathematical Physics vol 113 no2 pp 299ndash339 1987
[8] D Mundici ldquoFarey stellar subdivisions ultrasimplicial groupsand119870
0of AF119862-algebrasrdquoAdvances inMathematics vol 68 no
1 pp 23ndash39 1988[9] F P Boca ldquoAnAF algebra associatedwith the Farey tessellationrdquo
Canadian Journal of Mathematics vol 60 no 5 pp 975ndash10002008
[10] D Mundici ldquoRevisiting the Farey AF-algebrardquoMilan Journal ofMathematics vol 79 no 2 pp 643ndash656 2011
[11] D Mundici ldquoRecognizing the Farey-Stern-Brocot AF algebraDedicated to the memory of Renato Caccioppolirdquo RendicontiLincei-Matematica e Applicazioni vol 20 pp 327ndash338 2009
[12] C Eckhardt ldquoA noncommutative Gauss maprdquo MathematicaScandinavica vol 108 no 2 pp 233ndash250 2011
[13] G Panti ldquoPrime ideals in free 119897 -groups and free vector latticesrdquoJournal of Algebra vol 219 no 1 pp 173ndash200 1999
[14] E G Effros and C L Shen ldquoApproximately finite 119862-algebrasand continued fractionsrdquo Indiana University Mathematics Jour-nal vol 29 no 2 pp 191ndash204 1980
[15] W A Veech ldquoThe Teichmuller geodesic flowrdquo Annals ofMathematics Second Series vol 124 no 3 pp 441ndash530 1986
[16] I VNikolaev ldquoOn aTeichmuller functor between the categoriesof complex tori and the Effros-Shen algebrasrdquoNew York Journalof Mathematics vol 15 pp 125ndash132 2009
[17] I Nikolaev ldquoRiemann surfaces and AF-algebrasrdquo Annals ofFunctional Analysis vol 7 no 2 pp 371ndash380 2016
[18] I Nikolaev ldquoCluster Clowast-algebras and knot polynomialsrdquo httparxivorgabs160301180
[19] I Nikolaev ldquoK-theory of cluster Clowast-algebrasrdquo httparxivorgabs151200276
[20] L Bernstein The Jacobi-Perron AlgorithmmdashIts Theory andApplication vol 207 of Lecture Notes in Mathematics SpringerBerlin Germany 1971
[21] A Connes ldquoVon Neumann algebrasrdquo in Proceedings of theInternational Congress of Mathematicians (ICM rsquo78) pp 97ndash109Helsinki Finland 1978
[22] W P Thurston ldquoOn the geometry and dynamics of diffeo-morphisms of surfacesrdquo Bulletin of the American MathematicalSociety vol 19 no 2 pp 417ndash431 1988
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Journal of Function Spaces
References
[1] S Fomin and A Zelevinsky ldquoCluster algebras I foundationsrdquoJournal of the American Mathematical Society vol 15 no 2 pp497ndash529 2002
[2] S Fomin M Shapiro and D Thurston ldquoCluster algebras andtriangulated surfaces I Cluster complexesrdquo Acta Mathematicavol 201 no 1 pp 83ndash146 2008
[3] L K Williams ldquoCluster algebras an introductionrdquo Bulletin ofthe American Mathematical Society vol 51 no 1 pp 1ndash26 2014
[4] E G Effros Dimensions and Clowast-Algebras vol 46 of CBMSRegional Conference Series in Mathematics Conference Boardof the Mathematical Sciences 1981
[5] B Blackadar K-Theory for Operator Algebras vol 5 MSRIPublications Springer 1986
[6] O Bratteli ldquoInductive limits of finite dimensional Clowast-algebrasrdquoTransactions of the American Mathematical Society vol 171 pp195ndash234 1972
[7] R C Penner ldquoThe decorated Teichmuller space of puncturedsurfacesrdquoCommunications inMathematical Physics vol 113 no2 pp 299ndash339 1987
[8] D Mundici ldquoFarey stellar subdivisions ultrasimplicial groupsand119870
0of AF119862-algebrasrdquoAdvances inMathematics vol 68 no
1 pp 23ndash39 1988[9] F P Boca ldquoAnAF algebra associatedwith the Farey tessellationrdquo
Canadian Journal of Mathematics vol 60 no 5 pp 975ndash10002008
[10] D Mundici ldquoRevisiting the Farey AF-algebrardquoMilan Journal ofMathematics vol 79 no 2 pp 643ndash656 2011
[11] D Mundici ldquoRecognizing the Farey-Stern-Brocot AF algebraDedicated to the memory of Renato Caccioppolirdquo RendicontiLincei-Matematica e Applicazioni vol 20 pp 327ndash338 2009
[12] C Eckhardt ldquoA noncommutative Gauss maprdquo MathematicaScandinavica vol 108 no 2 pp 233ndash250 2011
[13] G Panti ldquoPrime ideals in free 119897 -groups and free vector latticesrdquoJournal of Algebra vol 219 no 1 pp 173ndash200 1999
[14] E G Effros and C L Shen ldquoApproximately finite 119862-algebrasand continued fractionsrdquo Indiana University Mathematics Jour-nal vol 29 no 2 pp 191ndash204 1980
[15] W A Veech ldquoThe Teichmuller geodesic flowrdquo Annals ofMathematics Second Series vol 124 no 3 pp 441ndash530 1986
[16] I VNikolaev ldquoOn aTeichmuller functor between the categoriesof complex tori and the Effros-Shen algebrasrdquoNew York Journalof Mathematics vol 15 pp 125ndash132 2009
[17] I Nikolaev ldquoRiemann surfaces and AF-algebrasrdquo Annals ofFunctional Analysis vol 7 no 2 pp 371ndash380 2016
[18] I Nikolaev ldquoCluster Clowast-algebras and knot polynomialsrdquo httparxivorgabs160301180
[19] I Nikolaev ldquoK-theory of cluster Clowast-algebrasrdquo httparxivorgabs151200276
[20] L Bernstein The Jacobi-Perron AlgorithmmdashIts Theory andApplication vol 207 of Lecture Notes in Mathematics SpringerBerlin Germany 1971
[21] A Connes ldquoVon Neumann algebrasrdquo in Proceedings of theInternational Congress of Mathematicians (ICM rsquo78) pp 97ndash109Helsinki Finland 1978
[22] W P Thurston ldquoOn the geometry and dynamics of diffeo-morphisms of surfacesrdquo Bulletin of the American MathematicalSociety vol 19 no 2 pp 417ndash431 1988
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of