s. paul smith · s. paul smith university of washington seattle, wa 98195....
TRANSCRIPT
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Graded modules over path algebras
S. Paul Smith
University of WashingtonSeattle, WA 98195.
April 13, 2012
ARTIN (Algebra and Ring Theory in the North)
Newcastle (UK)
S. Paul Smith Graded modules over path algebras
![Page 2: S. Paul Smith · S. Paul Smith University of Washington Seattle, WA 98195. smith@math.washington.edu April 13, 2012 ARTIN (Algebra and Ring Theory in the North) Newcastle (UK) S](https://reader036.vdocuments.us/reader036/viewer/2022090607/605d27b09615c93a73619af9/html5/thumbnails/2.jpg)
References for this talk
Category equivalences involving graded modules over pathalgebras of quivers, Adv. in Math., in press. arXiv:1107.3511
(with C. Holdaway) An equivalence of categories for gradedmodules over monomial algebras and path algebras of quivers,J. Algebra, 353 (2011) 249-260. arXiv: 1109.4387
“Degenerate” 3-dimensional Sklyanin algebras are monomialalgebras, J. Algebra, 358 (2012) 74-86. arXiv: 1112.5809
S. Paul Smith Graded modules over path algebras
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Path algebras
Let Q be a directed graph with finitely many arrows and vertices.Call Q a quiver.
Loops and multiple arrows between vertices are allowed, e.g.,
•
��~~~~~~~
��
•99 <<// •
__@@@@@@@
ee
A path is a concatenation of arrows.
S. Paul Smith Graded modules over path algebras
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Path algebras I
For example, reading from right to left, as with composition offunctions, p = cbgffeb is a path in
•
��~~~~~~~b
��
a
•99c<<
d //
e
•
__@@@@@@@fee
g
Fix a field k , usually unimportant.
The path algebra of Q is the k-algebra
kQ := the linear span of all paths in Q
with multiplication
pq :=
{pq if q ends where p begins
0 otherwise.
S. Paul Smith Graded modules over path algebras
![Page 5: S. Paul Smith · S. Paul Smith University of Washington Seattle, WA 98195. smith@math.washington.edu April 13, 2012 ARTIN (Algebra and Ring Theory in the North) Newcastle (UK) S](https://reader036.vdocuments.us/reader036/viewer/2022090607/605d27b09615c93a73619af9/html5/thumbnails/5.jpg)
Path algebras II
WriteI = the set of vertices in Q
andei = the empty path at vertex i , i ∈ I .
The ei s are mutually orthogonal idempotents.and ∑
i∈I
ei = 1,
the identity in kQ, because
eipej =
{p if p is a path from j to i
0 otherwise.
S. Paul Smith Graded modules over path algebras
![Page 6: S. Paul Smith · S. Paul Smith University of Washington Seattle, WA 98195. smith@math.washington.edu April 13, 2012 ARTIN (Algebra and Ring Theory in the North) Newcastle (UK) S](https://reader036.vdocuments.us/reader036/viewer/2022090607/605d27b09615c93a73619af9/html5/thumbnails/6.jpg)
Path algebras II
WriteI = the set of vertices in Q
andei = the empty path at vertex i , i ∈ I .
The ei s are mutually orthogonal idempotents.and ∑
i∈I
ei = 1,
the identity in kQ, because
eipej =
{p if p is a path from j to i
0 otherwise.
S. Paul Smith Graded modules over path algebras
![Page 7: S. Paul Smith · S. Paul Smith University of Washington Seattle, WA 98195. smith@math.washington.edu April 13, 2012 ARTIN (Algebra and Ring Theory in the North) Newcastle (UK) S](https://reader036.vdocuments.us/reader036/viewer/2022090607/605d27b09615c93a73619af9/html5/thumbnails/7.jpg)
Grading and QGr(kQ)
Make kQ a graded ring by declaring its degree-n component is
kQn := the linear span of length-n paths.
Define
QGr(kQ) :=Gr(kQ)
Fdim(kQ)=
Z-graded left kQ-modules
{M = sum of its finite diml submods}
S. Paul Smith Graded modules over path algebras
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Grading and QGr(kQ)
Make kQ a graded ring by declaring its degree-n component is
kQn := the linear span of length-n paths.
Define
QGr(kQ) :=Gr(kQ)
Fdim(kQ)=
Z-graded left kQ-modules
{M = sum of its finite diml submods}
S. Paul Smith Graded modules over path algebras
![Page 9: S. Paul Smith · S. Paul Smith University of Washington Seattle, WA 98195. smith@math.washington.edu April 13, 2012 ARTIN (Algebra and Ring Theory in the North) Newcastle (UK) S](https://reader036.vdocuments.us/reader036/viewer/2022090607/605d27b09615c93a73619af9/html5/thumbnails/9.jpg)
Philosophy of nc projective algebraic geometry
Certain abelian and triangulated categories behave as if they are“quasi-coherent sheaves” on “non-commutative schemes”.
Examples:
If R is a ring ModR is the category of “quasi-coherent sheaveson the non-commutative affine scheme Specnc R”.
QGr(kQ) is the category of “quasi-coherent sheaves on thenon-commutative projective scheme Projnc(kQ)”.
The geometric object is defined implicitly, cf., definition of stacks.
Today: just treat QGr(kQ) as an interesting abelian category.
S. Paul Smith Graded modules over path algebras
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Philosophy of nc projective algebraic geometry
Certain abelian and triangulated categories behave as if they are“quasi-coherent sheaves” on “non-commutative schemes”.
Examples:
If R is a ring ModR is the category of “quasi-coherent sheaveson the non-commutative affine scheme Specnc R”.
QGr(kQ) is the category of “quasi-coherent sheaves on thenon-commutative projective scheme Projnc(kQ)”.
The geometric object is defined implicitly, cf., definition of stacks.
Today: just treat QGr(kQ) as an interesting abelian category.
S. Paul Smith Graded modules over path algebras
![Page 11: S. Paul Smith · S. Paul Smith University of Washington Seattle, WA 98195. smith@math.washington.edu April 13, 2012 ARTIN (Algebra and Ring Theory in the North) Newcastle (UK) S](https://reader036.vdocuments.us/reader036/viewer/2022090607/605d27b09615c93a73619af9/html5/thumbnails/11.jpg)
Theorem (S)
There are category equivalences
QGr(kQ) ≡ ModS(Q) ≡ GrL(Q◦) ≡ ModL(Q◦)0 ≡ QGr(kQ(n)).
where
S(Q) = lim−→EndkI (kQ⊗n1 ) is a direct limit of finite dimensional
semisimple algebras;
Q◦ is the quiver without sources or sinks that is obtained byrepeatedly removing all sinks and sources from Q;
L(Q◦) is the Leavitt path algebra of Q◦;
L(Q◦)0 is its degree zero component;
Q(n) is the quiver whose incidence matrix is the nth power ofthat for Q.
All short exact sequences in qgr(kQ), the full subcategory offinitely presented objects in QGr(kQ), split.
S. Paul Smith Graded modules over path algebras
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Theorem (S)
There are category equivalences
QGr(kQ) ≡ ModS(Q) ≡ GrL(Q◦) ≡ ModL(Q◦)0 ≡ QGr(kQ(n)).
where
S(Q) = lim−→EndkI (kQ⊗n1 ) is a direct limit of finite dimensional
semisimple algebras;
Q◦ is the quiver without sources or sinks that is obtained byrepeatedly removing all sinks and sources from Q;
L(Q◦) is the Leavitt path algebra of Q◦;
L(Q◦)0 is its degree zero component;
Q(n) is the quiver whose incidence matrix is the nth power ofthat for Q.
All short exact sequences in qgr(kQ), the full subcategory offinitely presented objects in QGr(kQ), split.
S. Paul Smith Graded modules over path algebras
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Hazrat+Dade =⇒ GrL(Q◦) ≡ ModL(Q◦)0
Theorem (R. Hazrat, arXiv:1005.1900)
L(Q)nL(Q)−n = L(Q)0 for all n ∈ Z if and only if Q does not havea sink.
Theorem (Dade)
If A is a Z-graded ring such that AnA−n = A0 for all n ∈ Z, then
GrA ≡ ModA0
via M M0.
S. Paul Smith Graded modules over path algebras
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UHF example
Q := one vertex and n arrows.CQ = C〈x1, . . . , xn〉 = the free algebra.
S(Q) is the direct limit (in the category of C-algebras) of
Mn(C)→ Mn(C)⊗2 → Mn(C)⊗3 → · · ·
where all the maps are f 7→ 1⊗ f .
The direct limit in the category of C∗-algebras is
Mn∞(C) = a uniformly hyperfinite C∗-algebra,
a von Neumann algebra.
S(Q) is a dense subalgebra of Mn∞(C).Equivalently, Mn∞(C) is the norm closure of S(Q).
S. Paul Smith Graded modules over path algebras
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UHF example
Q := one vertex and n arrows.CQ = C〈x1, . . . , xn〉 = the free algebra.
S(Q) is the direct limit (in the category of C-algebras) of
Mn(C)→ Mn(C)⊗2 → Mn(C)⊗3 → · · ·
where all the maps are f 7→ 1⊗ f .
The direct limit in the category of C∗-algebras is
Mn∞(C) = a uniformly hyperfinite C∗-algebra,
a von Neumann algebra.
S(Q) is a dense subalgebra of Mn∞(C).Equivalently, Mn∞(C) is the norm closure of S(Q).
S. Paul Smith Graded modules over path algebras
![Page 16: S. Paul Smith · S. Paul Smith University of Washington Seattle, WA 98195. smith@math.washington.edu April 13, 2012 ARTIN (Algebra and Ring Theory in the North) Newcastle (UK) S](https://reader036.vdocuments.us/reader036/viewer/2022090607/605d27b09615c93a73619af9/html5/thumbnails/16.jpg)
UHF example
Q := one vertex and n arrows.CQ = C〈x1, . . . , xn〉 = the free algebra.
S(Q) is the direct limit (in the category of C-algebras) of
Mn(C)→ Mn(C)⊗2 → Mn(C)⊗3 → · · ·
where all the maps are f 7→ 1⊗ f .
The direct limit in the category of C∗-algebras is
Mn∞(C) = a uniformly hyperfinite C∗-algebra,
a von Neumann algebra.
S(Q) is a dense subalgebra of Mn∞(C).Equivalently, Mn∞(C) is the norm closure of S(Q).
S. Paul Smith Graded modules over path algebras
![Page 17: S. Paul Smith · S. Paul Smith University of Washington Seattle, WA 98195. smith@math.washington.edu April 13, 2012 ARTIN (Algebra and Ring Theory in the North) Newcastle (UK) S](https://reader036.vdocuments.us/reader036/viewer/2022090607/605d27b09615c93a73619af9/html5/thumbnails/17.jpg)
UHF example
Q := one vertex and n arrows.CQ = C〈x1, . . . , xn〉 = the free algebra.
S(Q) is the direct limit (in the category of C-algebras) of
Mn(C)→ Mn(C)⊗2 → Mn(C)⊗3 → · · ·
where all the maps are f 7→ 1⊗ f .
The direct limit in the category of C∗-algebras is
Mn∞(C) = a uniformly hyperfinite C∗-algebra,
a von Neumann algebra.
S(Q) is a dense subalgebra of Mn∞(C).Equivalently, Mn∞(C) is the norm closure of S(Q).
S. Paul Smith Graded modules over path algebras
![Page 18: S. Paul Smith · S. Paul Smith University of Washington Seattle, WA 98195. smith@math.washington.edu April 13, 2012 ARTIN (Algebra and Ring Theory in the North) Newcastle (UK) S](https://reader036.vdocuments.us/reader036/viewer/2022090607/605d27b09615c93a73619af9/html5/thumbnails/18.jpg)
UHF example
Q := one vertex and n arrows.CQ = C〈x1, . . . , xn〉 = the free algebra.
S(Q) is the direct limit (in the category of C-algebras) of
Mn(C)→ Mn(C)⊗2 → Mn(C)⊗3 → · · ·
where all the maps are f 7→ 1⊗ f .
The direct limit in the category of C∗-algebras is
Mn∞(C) = a uniformly hyperfinite C∗-algebra,
a von Neumann algebra.
S(Q) is a dense subalgebra of Mn∞(C).Equivalently, Mn∞(C) is the norm closure of S(Q).
S. Paul Smith Graded modules over path algebras
![Page 19: S. Paul Smith · S. Paul Smith University of Washington Seattle, WA 98195. smith@math.washington.edu April 13, 2012 ARTIN (Algebra and Ring Theory in the North) Newcastle (UK) S](https://reader036.vdocuments.us/reader036/viewer/2022090607/605d27b09615c93a73619af9/html5/thumbnails/19.jpg)
The CAR (canonical anti-commutation relations) algebra
M2∞(C) is important in quantum mechanics
Theorem/Definition: Let B(H) be the bounded operators on aninfinite dimensional separable Hilbert space H. Let α : H → B(H)be any continuous linear map such that
α(g)α(h) + α(h)α(g) = 0 and
α(g)∗α(h) + α(h)α(g)∗ = 〈h, g〉 idH for all g , h ∈ H.
The C∗-subalgebra of B(H) generated by {α(h) | h ∈ H} isisomorphic to M2∞(C) and is called the CAR algebra.
C(
•99 ee)
= C〈x , y〉 QGr M2∞(C)
S. Paul Smith Graded modules over path algebras
![Page 20: S. Paul Smith · S. Paul Smith University of Washington Seattle, WA 98195. smith@math.washington.edu April 13, 2012 ARTIN (Algebra and Ring Theory in the North) Newcastle (UK) S](https://reader036.vdocuments.us/reader036/viewer/2022090607/605d27b09615c93a73619af9/html5/thumbnails/20.jpg)
The CAR (canonical anti-commutation relations) algebra
M2∞(C) is important in quantum mechanics
Theorem/Definition: Let B(H) be the bounded operators on aninfinite dimensional separable Hilbert space H. Let α : H → B(H)be any continuous linear map such that
α(g)α(h) + α(h)α(g) = 0 and
α(g)∗α(h) + α(h)α(g)∗ = 〈h, g〉 idH for all g , h ∈ H.
The C∗-subalgebra of B(H) generated by {α(h) | h ∈ H} isisomorphic to M2∞(C) and is called the CAR algebra.
C(
•99 ee)
= C〈x , y〉 QGr M2∞(C)
S. Paul Smith Graded modules over path algebras
![Page 21: S. Paul Smith · S. Paul Smith University of Washington Seattle, WA 98195. smith@math.washington.edu April 13, 2012 ARTIN (Algebra and Ring Theory in the North) Newcastle (UK) S](https://reader036.vdocuments.us/reader036/viewer/2022090607/605d27b09615c93a73619af9/html5/thumbnails/21.jpg)
The CAR (canonical anti-commutation relations) algebra
M2∞(C) is important in quantum mechanics
Theorem/Definition: Let B(H) be the bounded operators on aninfinite dimensional separable Hilbert space H. Let α : H → B(H)be any continuous linear map such that
α(g)α(h) + α(h)α(g) = 0 and
α(g)∗α(h) + α(h)α(g)∗ = 〈h, g〉 idH for all g , h ∈ H.
The C∗-subalgebra of B(H) generated by {α(h) | h ∈ H} isisomorphic to M2∞(C) and is called the CAR algebra.
C(
•99 ee)
= C〈x , y〉 QGr M2∞(C)
S. Paul Smith Graded modules over path algebras
![Page 22: S. Paul Smith · S. Paul Smith University of Washington Seattle, WA 98195. smith@math.washington.edu April 13, 2012 ARTIN (Algebra and Ring Theory in the North) Newcastle (UK) S](https://reader036.vdocuments.us/reader036/viewer/2022090607/605d27b09615c93a73619af9/html5/thumbnails/22.jpg)
The CAR (canonical anti-commutation relations) algebra
M2∞(C) is important in quantum mechanics
Theorem/Definition: Let B(H) be the bounded operators on aninfinite dimensional separable Hilbert space H. Let α : H → B(H)be any continuous linear map such that
α(g)α(h) + α(h)α(g) = 0 and
α(g)∗α(h) + α(h)α(g)∗ = 〈h, g〉 idH for all g , h ∈ H.
The C∗-subalgebra of B(H) generated by {α(h) | h ∈ H} isisomorphic to M2∞(C) and is called the CAR algebra.
C(
•99 ee)
= C〈x , y〉 QGr M2∞(C)
S. Paul Smith Graded modules over path algebras
![Page 23: S. Paul Smith · S. Paul Smith University of Washington Seattle, WA 98195. smith@math.washington.edu April 13, 2012 ARTIN (Algebra and Ring Theory in the North) Newcastle (UK) S](https://reader036.vdocuments.us/reader036/viewer/2022090607/605d27b09615c93a73619af9/html5/thumbnails/23.jpg)
The Veronese subalgebra kQ(m)
Vertices in Q(m) = the vertices in Q
Arrows in Q(m) = the paths of length m in Q.
Corollary
QGr(kQ) ≡ QGr(kQ(m))
Proof: S(Q(m)) = lim−→nEndkI kQmn is a subsystem of the directed
system defining S(Q) so
S(Q) = S(Q(m)).
S. Paul Smith Graded modules over path algebras
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The Veronese subalgebra kQ(m)
Vertices in Q(m) = the vertices in Q
Arrows in Q(m) = the paths of length m in Q.
Corollary
QGr(kQ) ≡ QGr(kQ(m))
Proof: S(Q(m)) = lim−→nEndkI kQmn is a subsystem of the directed
system defining S(Q) so
S(Q) = S(Q(m)).
S. Paul Smith Graded modules over path algebras
![Page 25: S. Paul Smith · S. Paul Smith University of Washington Seattle, WA 98195. smith@math.washington.edu April 13, 2012 ARTIN (Algebra and Ring Theory in the North) Newcastle (UK) S](https://reader036.vdocuments.us/reader036/viewer/2022090607/605d27b09615c93a73619af9/html5/thumbnails/25.jpg)
An example by Raf Bocklandt
Q := vertices the elements of the group Z/n and arrows i → i + 1for all i . There is a unique path of length n starting at each i andthat path ends at i , so Q(n) is
•��
· · · •��
n vertices
and kQ(n) ∼= k[x ]⊕n. Now,
Proj(k[x ]⊕n
)= n points = Spec(k⊕n)
so QGr(kQ) ≡ QGr(kQ(n)) ≡ QGr(k[x ]⊕n
)but, using Serre [FAC],
QGr(k[x ]⊕n
)≡Qcoh
(Proj
(k[x ]⊕n
))≡ Qcoh(n points) ≡ Mod(k⊕n).
It is the case that S(Q) ∼= k⊕n.
S. Paul Smith Graded modules over path algebras
![Page 26: S. Paul Smith · S. Paul Smith University of Washington Seattle, WA 98195. smith@math.washington.edu April 13, 2012 ARTIN (Algebra and Ring Theory in the North) Newcastle (UK) S](https://reader036.vdocuments.us/reader036/viewer/2022090607/605d27b09615c93a73619af9/html5/thumbnails/26.jpg)
An example by Raf Bocklandt
Q := vertices the elements of the group Z/n and arrows i → i + 1for all i . There is a unique path of length n starting at each i andthat path ends at i , so Q(n) is
•��
· · · •��
n vertices
and kQ(n) ∼= k[x ]⊕n. Now,
Proj(k[x ]⊕n
)= n points = Spec(k⊕n)
so QGr(kQ) ≡ QGr(kQ(n)) ≡ QGr(k[x ]⊕n
)but, using Serre [FAC],
QGr(k[x ]⊕n
)≡Qcoh
(Proj
(k[x ]⊕n
))≡ Qcoh(n points) ≡ Mod(k⊕n).
It is the case that S(Q) ∼= k⊕n.
S. Paul Smith Graded modules over path algebras
![Page 27: S. Paul Smith · S. Paul Smith University of Washington Seattle, WA 98195. smith@math.washington.edu April 13, 2012 ARTIN (Algebra and Ring Theory in the North) Newcastle (UK) S](https://reader036.vdocuments.us/reader036/viewer/2022090607/605d27b09615c93a73619af9/html5/thumbnails/27.jpg)
An example by Raf Bocklandt
Q := vertices the elements of the group Z/n and arrows i → i + 1for all i . There is a unique path of length n starting at each i andthat path ends at i , so Q(n) is
•��
· · · •��
n vertices
and kQ(n) ∼= k[x ]⊕n. Now,
Proj(k[x ]⊕n
)= n points = Spec(k⊕n)
so QGr(kQ) ≡ QGr(kQ(n)) ≡ QGr(k[x ]⊕n
)but, using Serre [FAC],
QGr(k[x ]⊕n
)≡Qcoh
(Proj
(k[x ]⊕n
))≡ Qcoh(n points) ≡ Mod(k⊕n).
It is the case that S(Q) ∼= k⊕n.
S. Paul Smith Graded modules over path algebras
![Page 28: S. Paul Smith · S. Paul Smith University of Washington Seattle, WA 98195. smith@math.washington.edu April 13, 2012 ARTIN (Algebra and Ring Theory in the North) Newcastle (UK) S](https://reader036.vdocuments.us/reader036/viewer/2022090607/605d27b09615c93a73619af9/html5/thumbnails/28.jpg)
An example by Raf Bocklandt
Q := vertices the elements of the group Z/n and arrows i → i + 1for all i . There is a unique path of length n starting at each i andthat path ends at i , so Q(n) is
•��
· · · •��
n vertices
and kQ(n) ∼= k[x ]⊕n. Now,
Proj(k[x ]⊕n
)= n points = Spec(k⊕n)
so QGr(kQ) ≡ QGr(kQ(n)) ≡ QGr(k[x ]⊕n
)but, using Serre [FAC],
QGr(k[x ]⊕n
)≡Qcoh
(Proj
(k[x ]⊕n
))≡ Qcoh(n points) ≡ Mod(k⊕n).
It is the case that S(Q) ∼= k⊕n.
S. Paul Smith Graded modules over path algebras
![Page 29: S. Paul Smith · S. Paul Smith University of Washington Seattle, WA 98195. smith@math.washington.edu April 13, 2012 ARTIN (Algebra and Ring Theory in the North) Newcastle (UK) S](https://reader036.vdocuments.us/reader036/viewer/2022090607/605d27b09615c93a73619af9/html5/thumbnails/29.jpg)
An example by Raf Bocklandt
Q := vertices the elements of the group Z/n and arrows i → i + 1for all i . There is a unique path of length n starting at each i andthat path ends at i , so Q(n) is
•��
· · · •��
n vertices
and kQ(n) ∼= k[x ]⊕n. Now,
Proj(k[x ]⊕n
)= n points = Spec(k⊕n)
so QGr(kQ) ≡ QGr(kQ(n)) ≡ QGr(k[x ]⊕n
)but, using Serre [FAC],
QGr(k[x ]⊕n
)≡Qcoh
(Proj
(k[x ]⊕n
))≡ Qcoh(n points) ≡ Mod(k⊕n).
It is the case that S(Q) ∼= k⊕n.
S. Paul Smith Graded modules over path algebras
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An example by Raf Bocklandt
Q := vertices the elements of the group Z/n and arrows i → i + 1for all i . There is a unique path of length n starting at each i andthat path ends at i , so Q(n) is
•��
· · · •��
n vertices
and kQ(n) ∼= k[x ]⊕n. Now,
Proj(k[x ]⊕n
)= n points = Spec(k⊕n)
so QGr(kQ) ≡ QGr(kQ(n)) ≡ QGr(k[x ]⊕n
)but, using Serre [FAC],
QGr(k[x ]⊕n
)≡Qcoh
(Proj
(k[x ]⊕n
))≡ Qcoh(n points) ≡ Mod(k⊕n).
It is the case that S(Q) ∼= k⊕n.
S. Paul Smith Graded modules over path algebras
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Make qgr(kQ) a triangulated category
S(Q) is von Neumann regular
all short exact sequences in modS(Q) split
all short exact sequences in qgr(kQ) split
Make qgr(kQ) a triangulated category:
suspension functor Σ = the Serre degree twist (−1)
declare that the distinguished triangles in qgr(kQ) are alldirect sums of the following triangles:
M→ 0→ ΣM id−→ ΣM,
M id−→M→ 0→ ΣM,
0→M id−→M→ 0,
S. Paul Smith Graded modules over path algebras
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Make qgr(kQ) a triangulated category
S(Q) is von Neumann regular
all short exact sequences in modS(Q) split
all short exact sequences in qgr(kQ) split
Make qgr(kQ) a triangulated category:
suspension functor Σ = the Serre degree twist (−1)
declare that the distinguished triangles in qgr(kQ) are alldirect sums of the following triangles:
M→ 0→ ΣM id−→ ΣM,
M id−→M→ 0→ ΣM,
0→M id−→M→ 0,
S. Paul Smith Graded modules over path algebras
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The singularity category of a finite dimensional algebra
Λ = a finite dimensional algebramod(Λ) = finite dimensional left Λ-modulesDb(modΛ) = the bounded derived categoryDperf(modΛ) = the full subcategory of perfect complexes
Dsing(Λ) :=Db(modΛ)
Dperf(modΛ)
S. Paul Smith Graded modules over path algebras
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The singularity category of a radical-square-zero algebra
Theorem (X.-W. Chen)
Let Λ be a finite dimensional k-algebra and J its Jacobson radical.Suppose J2 = 0. Define
S(Λ) := lim−→EndΛ(J⊗n)
where the maps in the directed system are f 7→ idJ ⊗f and
B := lim−→HomΛ(J⊗n, J⊗n−1).
Then
B is an invertible S(Λ)-bimodule
J is a progenerator in Dsg(Λ) with endomorphism ring S(Λ);
HomDsing(Λ)(J,−) is an equivalence of triangulated categories(Dsing(Λ), [1]
)≡(modS(Λ),−⊗S(Λ) B
)where proj S(Λ) is the category of finitely presented rightS(Λ)-modules.
S. Paul Smith Graded modules over path algebras
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Chen + S
Let Λ = kQ/kQ≥2.
Easy: S(Λ) = S(Q).
Theorem (Chen + S)
qgr(kQ) ≡ modS(Q) = modS(Λ) ≡ Dsing(Λ)
as triangulated categories.
S. Paul Smith Graded modules over path algebras
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Chen + S
Let Λ = kQ/kQ≥2.
Easy: S(Λ) = S(Q).
Theorem (Chen + S)
qgr(kQ) ≡ modS(Q) = modS(Λ) ≡ Dsing(Λ)
as triangulated categories.
S. Paul Smith Graded modules over path algebras
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Chen + S
Let Λ = kQ/kQ≥2.
Easy: S(Λ) = S(Q).
Theorem (Chen + S)
qgr(kQ) ≡ modS(Q) = modS(Λ) ≡ Dsing(Λ)
as triangulated categories.
S. Paul Smith Graded modules over path algebras
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Monomial algebras
A monomial algebra is an algebra of the form
A =k〈x1, . . . , xg 〉(w1, . . . ,wr )
(1)
where w1, . . . ,wr are words in the letters x1, . . . , xg .
Theorem (Holdaway-S)
Let A be a monomial algebra and Q its Ufnarovskii graph. There isa homomorphism of graded algebras f : A→ kQ such thatkQ ⊗A − induces an equivalence of categories
QGr A ≡ QGr kQ.
More general monomial algebras: This theorem also holds whenA = the path algebra of a quiver modulo a finite number ofrelations of the form “path= 0”.
S. Paul Smith Graded modules over path algebras
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Monomial algebras
A monomial algebra is an algebra of the form
A =k〈x1, . . . , xg 〉(w1, . . . ,wr )
(1)
where w1, . . . ,wr are words in the letters x1, . . . , xg .
Theorem (Holdaway-S)
Let A be a monomial algebra and Q its Ufnarovskii graph. There isa homomorphism of graded algebras f : A→ kQ such thatkQ ⊗A − induces an equivalence of categories
QGr A ≡ QGr kQ.
More general monomial algebras: This theorem also holds whenA = the path algebra of a quiver modulo a finite number ofrelations of the form “path= 0”.
S. Paul Smith Graded modules over path algebras
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Sklyanin algebras
If (a, b, c) ∈ P2 define Sa,b,c =k〈x , y , z〉(f1, f2, f3)
where
f1 =ayz + bzy + cx2
f2 =azx + bxz + cy 2
f3 =axy + byx + cz2.
Call Sa,b,c a 3-dimensional Sklyanin algebra if (a, b, c) ∈ P2 −D
where
D :={
(1, 0, 0), (0, 1, 0), (0, 0, 1)}t{
(a, b, c) | a3 = b3 = c3}
⊂ P2k .
Artin, Tate, and Van den Bergh:
Sa,b,c is like the polynomial ring k[X ,Y ,Z ]
QGr Sa,b,c is like QcohP2.
S. Paul Smith Graded modules over path algebras
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“Degenerate” Sklyanin algebras: assume (a, b, c) ∈ D
Theorem (C. Walton)
Sa,b,c has infinite global dimension, is not noetherian, hasexponential growth, and has zero divisors.
Theorem (S)
Let k be a field having a primitive cube root of unity ω.
1 If a = b, then
Sa,b,c∼=
k〈u, v ,w〉(u2, v 2,w 2)
.
2 If a 6= b, then
Sa,b,c∼=
k〈u, v ,w〉(uv , vw ,wu)
.
3 Gr(Sa,b,c) ≡ Gr(Sa′,b′,c ′) for all (a, b, c), (a′, b′, c ′) ∈ D
S. Paul Smith Graded modules over path algebras
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“Degenerate” Sklyanin algebras: assume (a, b, c) ∈ D
Theorem (S)
There is a quiver Q, independent of (a, b, c) ∈ D, such that
QGr(Sa,b,c) ≡ QGr kQ ≡ ModS(Q)
where S(Q) = lim−→ Sn and each Sn is a product of three matrix
algebras. There is an action of µ3 = 3√
1 ⊂ k× as automorphismsof the free algebra F = k〈X ,Y 〉 such that
QGr(Sa,b,c) ≡ QGr(F o µ3).
S. Paul Smith Graded modules over path algebras
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“Degenerate” Sklyanin algebras: assume (a, b, c) ∈ D
Theorem (S)
There is a quiver Q, independent of (a, b, c) ∈ D, such that
QGr(Sa,b,c) ≡ QGr kQ ≡ ModS(Q)
where S(Q) = lim−→ Sn and each Sn is a product of three matrix
algebras. There is an action of µ3 = 3√
1 ⊂ k× as automorphismsof the free algebra F = k〈X ,Y 〉 such that
QGr(Sa,b,c) ≡ QGr(F o µ3).
S. Paul Smith Graded modules over path algebras
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“Degenerate” Sklyanin algebras: assume (a, b, c) ∈ D
The Ufnarovskii graph for A = k〈u, v ,w〉/(u2, v 2,w 2) is the quiver
•u1
@@@
��@@@
u2
��•
w1~~~
??~~~
w2
<< •v1
oo
v2qq (2)
The equivalence QGr A ≡ QGr kQ is induced by the k-algebrahomomorphism f : A→ kQ defined by
f (u) = u1 + u2,
f (v) = v1 + v2,
f (w) = w1 + w2.
S. Paul Smith Graded modules over path algebras
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“Degenerate” Sklyanin algebras: assume (a, b, c) ∈ D
The Ufnarovskii graph for A′ = k〈u, v ,w〉/(uv , vw ,wu) is thequiver
•u2
��~~~~~~~
u1
��
•w1 99 w2
// •
v2
__@@@@@@@v1ee
(3)
The equivalence QGr A′ ≡ QGr kQ ′ is induced by the k-algebrahomomorphism f ′ : A′ → kQ ′ defined by
f ′(u) = u1 + u2,
f ′(v) = v1 + v2,
f ′(w) = w1 + w2.
S. Paul Smith Graded modules over path algebras
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“Degenerate” Sklyanin algebras: assume (a, b, c) ∈ D
The 3-Veronese quivers of Q and Q ′ are the same, then QGr kQ isequivalent to QGr kQ ′.The incidence matrix of the nth Veronese quiver is the nth power ofthe incidence matrix of the original quiver. The incidence matricesof Q and Q ′ are0 1 1
1 0 11 1 0
and
1 1 00 1 11 0 1
and the third power of each is2 3 3
3 2 33 3 2
so QGr kQ ≡ QGr kQ ′.
S. Paul Smith Graded modules over path algebras
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Examples of interest to C∗-algebraists
Take C = the base field.
S(Q) = lim−→EndCI (CQn) in the category of C-algebras.
S(Q) = lim−→EndCI (CQn) in the category of C∗-algebras.
S(Q) is a dense subalgebra of S(Q).
S(Q) is an AF algebra. (AF=almost finite dimensional)AF-algebras are an important class of C∗-algebras.Some interesting AF-algebras arise as S(Q):
Mn∞(C)
Penrose tilings
Compact operators + multiples of the identity
S. Paul Smith Graded modules over path algebras
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Examples of interest to C∗-algebraists
Take C = the base field.
S(Q) = lim−→EndCI (CQn) in the category of C-algebras.
S(Q) = lim−→EndCI (CQn) in the category of C∗-algebras.
S(Q) is a dense subalgebra of S(Q).
S(Q) is an AF algebra. (AF=almost finite dimensional)AF-algebras are an important class of C∗-algebras.Some interesting AF-algebras arise as S(Q):
Mn∞(C)
Penrose tilings
Compact operators + multiples of the identity
S. Paul Smith Graded modules over path algebras
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Examples of interest to C∗-algebraists
Take C = the base field.
S(Q) = lim−→EndCI (CQn) in the category of C-algebras.
S(Q) = lim−→EndCI (CQn) in the category of C∗-algebras.
S(Q) is a dense subalgebra of S(Q).
S(Q) is an AF algebra. (AF=almost finite dimensional)AF-algebras are an important class of C∗-algebras.Some interesting AF-algebras arise as S(Q):
Mn∞(C)
Penrose tilings
Compact operators + multiples of the identity
S. Paul Smith Graded modules over path algebras
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Examples of interest to C∗-algebraists
Take C = the base field.
S(Q) = lim−→EndCI (CQn) in the category of C-algebras.
S(Q) = lim−→EndCI (CQn) in the category of C∗-algebras.
S(Q) is a dense subalgebra of S(Q).
S(Q) is an AF algebra. (AF=almost finite dimensional)AF-algebras are an important class of C∗-algebras.Some interesting AF-algebras arise as S(Q):
Mn∞(C)
Penrose tilings
Compact operators + multiples of the identity
S. Paul Smith Graded modules over path algebras
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Penrose tiling example
Connes associates to the space of Penrose tilings of R2 theC∗-algebra with Bratteli diagram
1 //
��>>>>>>>> 2 //
��>>>>>>>> 3 //
��>>>>>>>> 5 //
��>>>>>>>> 8 //
AAAAAAAA · · ·
1
@@��������1
@@��������2
@@��������3
@@��������5
>>}}}}}}}}· · ·
This is also the Bratteli diagram for S(Q) where
Q = • **99 •jj
ModS(Q) ≡ QGr CQ ≡ QGrC〈x , y〉
(y 2).
S. Paul Smith Graded modules over path algebras
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Compact operators + scalars
K(H) = the C∗-algebra of compact operators on an infinitedimensional separable Hilbert space H.
Let Q = •$$
•oozz
The Bratteli diagram for S(Q) is
1 //
��>>>>>>>> 1 //
��>>>>>>>> 1 //
��>>>>>>>> 1 //
��>>>>>>>> 1 //
AAAAAAAA · · ·
1 // 2 // 3 // 4 // 5 // · · ·
(4)
soS(Q) ∼= K(H)⊕ C idH .
S. Paul Smith Graded modules over path algebras
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Annoying pop-up
Projnc
(• %%
99 •ee
)is the space of Penrose tilings.
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173 of those visitors are me and my family
··_
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THE END
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Annoying pop-up
Projnc
(• %%
99 •ee
)is the space of Penrose tilings.
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Be the 263rd visitor to my blog, please.
173 of those visitors are me and my family
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THE END
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Annoying pop-up
Projnc
(• %%
99 •ee
)is the space of Penrose tilings.
http://www.ncaglife.wordpress.com/
Be the 263rd visitor to my blog, please.
173 of those visitors are me and my family
··_
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THE END
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Annoying pop-up
Projnc
(• %%
99 •ee
)is the space of Penrose tilings.
http://www.ncaglife.wordpress.com/
Be the 263rd visitor to my blog, please.
173 of those visitors are me and my family
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