KP MAPS AND CONTEXT-FREE LANGUAGES
Adam [email protected]
University of Warmia and Mazury, Olsztyn, Poland
SYMMETRIES AND INTEGRABILITY OF DIFFERENCE EQUATIONS XIII
KYUSHU UNIVERSITY, FUKUOKA, JAPANNOVEMBER 11–17, 2018
The first part jointly with Masatoshi Noumi
ADAM DOLIWA (UWM PL) KP MAPS AND CF LANGUAGES 11-17.11.2018 1 / 21
OUTLINE
1 NON-COMMUTATIVE DISCRETE KP SYSTEM AND YB MAPS
2 EXCURSION INTO CONTEXT-FREE LANGUAGES
3 GENERALIZED NON-COMMUTATIVE SYMMETRIC FUNCTIONS
ADAM DOLIWA (UWM PL) KP MAPS AND CF LANGUAGES 11-17.11.2018 2 / 21
OUTLINE
1 NON-COMMUTATIVE DISCRETE KP SYSTEM AND YB MAPS
2 EXCURSION INTO CONTEXT-FREE LANGUAGES
3 GENERALIZED NON-COMMUTATIVE SYMMETRIC FUNCTIONS
ADAM DOLIWA (UWM PL) KP MAPS AND CF LANGUAGES 11-17.11.2018 3 / 21
THE DISCRETE NON-COMMUTATIVE KP HIERARCHY
Consider the linear problem
Ψk+1 −Ψk(i) = Ψk ui,k , i = 1, . . . ,N, k ∈ Z
[Kajiwara, Noumi, Yamada 2002]NOTATION: Ψk(i)(n1, . . . , ni , . . . , nN) = Ψk (n1, . . . , ni + 1, . . . , nN)
The compatibility conditions give equations
uj,k ui,k(j) = ui,k uj,k(i), ui,k(j) + uj,k+1 = uj,k(i) + ui,k+1
The system is equivalent to the non-Abelian Hirota–Miwa system by [Nimmo 2006]
THEOREM
The non-commutative KP map
ui,k(j) = (ui,k − uj,k )−1ui,k (ui,k+1 − uj,k+1), 1 ≤ i 6= j ≤ N,
is multidimensionally consistent
ADAM DOLIWA (UWM PL) KP MAPS AND CF LANGUAGES 11-17.11.2018 4 / 21
FROM KP MAP TO YANG-BAXTER MAP
uj(i)
i(j)u
uj(l)
l
j
i
uj
luul(i)
ui
uj(i)u j
i
j
u i(j)
ul(j)
ui(l)
iu
ui(jl)
u i = (ui,k ), k ∈ Z or k ∈ ZP , ui,k+P = ui,k
=
1
2
3R23
R13
R12
1
2
3
12R
R23
R13
Ry~
~x
y
x
OBSERVATION [Adler, Bobenko, Suris 2004]
The COMPANION MAP R of a reversible three-dimensionally consistent map satisfiesset-theoretical Yang–Baxter equation
R12 ◦ R13 ◦ R23 = R23 ◦ R13 ◦ R12
ADAM DOLIWA (UWM PL) KP MAPS AND CF LANGUAGES 11-17.11.2018 5 / 21
THE COMPANION MAP
In terms of the companion variables
xk = ui,k , yk = uj,k(i), xk = uj,k , yk = ui,k(j)
the NC discrete KP system (in periodic reduction) reads
xk yk = yk xk , yk + xk+1 = xk + yk+1, k = 1, 2, . . . ,P
LEMMA
The auxiliary functions hk defined by
xk = yk − h−1k or equivalently by yk+1 = xk+1 + h−1
k
satify the systemyk hk = 1 + hk−1xk , k mod P
whose solution (by successive approximation technique starting from h(0)k = 0) is
hk = y−1k
(1 + y−1
k−1xk + y−1k−1y−1
k−2xk−1xk + y−1k−1y−1
k−2y−1k−3xk−2xk−1xk + . . .
)By construction R is "involutive"/invertible, i.e. R21 ◦ R12 = id
ADAM DOLIWA (UWM PL) KP MAPS AND CF LANGUAGES 11-17.11.2018 6 / 21
KP LANGUAGES AND THEIR CHARACTERISTIC SERIES
Given alphabet A = {a1, b1, . . . , aP , bP}, consider the system
Zk = 1 + ak Zk+1bk , k mod P
Given integer n ≥ 0, denote
[ab]0k = 1, [ab]n
k = ak [ab]n−1k+1 bk , with subscripts modulo P
EXAMPLE
For P = 3, k = 1 and n = 5 we have [ab]51 = a1a2a3a1a2b2b1b3b2b1
Zk =∑n≥0
[ab]nk = 1 + ak bk + ak ak+1bk+1bk + . . .
The set LP(k) = {[ab]nk , n ≥ 0} can be called a KP language
ADAM DOLIWA (UWM PL) KP MAPS AND CF LANGUAGES 11-17.11.2018 7 / 21
INVERSE OF THE CHARACTERISTIC SERIES OF THE KP LANGUAGE
COMPOSITION α |= n is a finite sequence α = (α1, α2, . . . , αm) of positive integers withn = α1 + α2 + · · ·+ αm, the length of the composition is |α| = m. Define
[ab]αk = [ab]α1k . . . [ab]αm
k .
PROPOSITION
The inverse of the series Zk =∑
n≥0[ab]nk is given by
Z−1k =
∑n≥0
∑α|=n
(−1)|α|[ab]αk = 1− ak bk + ak bk ak bk − ak ak+1bk+1bK + . . . .
Proof: Notice that Z−1k =
∑n≥0 cn, where cn is homogeneous polynomial of degree 2n
Z−1k Zk = 1 ⇒ c0 = 1, c0[ab]n
k + c1[ab]n−1k + · · ·+ cn = 0, n > 0
The identificationcn =
∑α|=n
(−1)|α|[ab]αk
follows from separation of the last component αm from the composition α∑α|=n
(−1)|α|[ab]αk = −n∑
i=1
∑β|=n−i
(−1)|β|[ab]βk [ab]ik .
ADAM DOLIWA (UWM PL) KP MAPS AND CF LANGUAGES 11-17.11.2018 8 / 21
OUTLINE
1 NON-COMMUTATIVE DISCRETE KP SYSTEM AND YB MAPS
2 EXCURSION INTO CONTEXT-FREE LANGUAGES
3 GENERALIZED NON-COMMUTATIVE SYMMETRIC FUNCTIONS
ADAM DOLIWA (UWM PL) KP MAPS AND CF LANGUAGES 11-17.11.2018 9 / 21
FORMAL LANGUAGES, GRAMMARS, ABSTRACT COMPUTING MACHINES
A = {a, b, c, . . . }— finite set (alphabet)A∗ — set of finite sequences (words) over A including the empty sequence λA∗ ⊃ L— formal languageA FORMAL GRAMMAR — set of production rules together with some auxiliary symbolswhich allow to construct words of a given language
A grammar (or rewriting system) G(N,A,R,S) is given by1 an alphabet N whose elements are called variables or nonterminals,2 an alphabet A (disjoint from N) whose elements are called terminals,3 a finite set R of rewritting rules or productions, each rule being a pair
(N ∪ A)∗N(N ∪ A)∗ → (A ∪ N)∗
4 an element S ∈ N called the initial variable (axiom)
A CONTEXT-FREE GRAMMAR — the left-hand side of each production rule consists ofonly a single auxiliary symbol
N → (A ∪ N)∗
ADAM DOLIWA (UWM PL) KP MAPS AND CF LANGUAGES 11-17.11.2018 10 / 21
THE CHOMSKY HIERARCHY
EXAMPLE
A = {a, b}, L = {λ, ab, aabb, aaabbb, . . . } = {anbn}n∈N0
Productions (here S is an auxiliary start symbol)
(1) S → λ, (2) S → aSb
derivation of aabb: S(2)⇒ aSb
(2)⇒ aaSbb(1)⇒ aabb
type language abstract machine (acceptor)0 recursively enumerable Turing machine1 context-sensitive non-deterministic linear-bounded TM2 context-free non-deterministic push-down automaton3 regular finite state automaton
The context-sensitive grammar G where N = {S,B}, A = {a, b, c}, S – the startsymbol, and productions:
1. S → aBSc, 2. S → abc
3. Ba→ aB, 4. Bb → bb
defines the language L(G) = {anbncn | n ≥ 1}ADAM DOLIWA (UWM PL) KP MAPS AND CF LANGUAGES 11-17.11.2018 11 / 21
FORMAL LANGUAGES AND FORMAL SERIES
EXAMPLE (CONTINUED)The CHARACTERISTIC SERIES Σ = λ+ ab + aabb + aaabbb + . . . of the language Lsatisfies equation
Σ = λ+ aΣb
1
2
3
a1
b1
b a
a
b3
3
22
EXAMPLE (THE CYCLIC REGULAR LANGUAGE C3)The characteristic series Σ1 = λ+ a1b1 + b3a3 + a1a2a3 + b3b2b1 + . . . of the regularlanguage accepted by the above finite state cyclic automaton satisfies the linear systemΣ1
Σ2
Σ3
=
λ00
+
0 a1 b3
b1 0 a2
a3 b2 0
Σ1
Σ2
Σ3
ADAM DOLIWA (UWM PL) KP MAPS AND CF LANGUAGES 11-17.11.2018 12 / 21
THE TRIAD: PROBLEM — TOOLS — SYMMETRY
SYMMETRY
TOOLSPROBLEM
ADAM DOLIWA (UWM PL) KP MAPS AND CF LANGUAGES 11-17.11.2018 13 / 21
THE TRIAD: PROBLEM — TOOLS — SYMMETRY
POLYNOMIAL EQUATIONS
FINITE GROUPS
RADICALS
ALGEBRAIC OPERATIONS
A polynomial equation can be solved using algebraic operations and radicals iff itsGalois group is solvable
ADAM DOLIWA (UWM PL) KP MAPS AND CF LANGUAGES 11-17.11.2018 14 / 21
THE TRIAD FOR REGULAR LANGUAGES
KleeneSchützenbergerReutenauer
REGULAR LANGUAGES
FREE HOPF ALGEBRAS
FINITE AUTOMATA
A language L ⊂ A∗ over finite alphabet A is regular iff its characteristic series belongsto the Sweedler’s dual of the free Hopf algebra Q〈A〉
ADAM DOLIWA (UWM PL) KP MAPS AND CF LANGUAGES 11-17.11.2018 15 / 21
OUTLINE
1 NON-COMMUTATIVE DISCRETE KP SYSTEM AND YB MAPS
2 EXCURSION INTO CONTEXT-FREE LANGUAGES
3 GENERALIZED NON-COMMUTATIVE SYMMETRIC FUNCTIONS
ADAM DOLIWA (UWM PL) KP MAPS AND CF LANGUAGES 11-17.11.2018 16 / 21
THE ALGEBRA OF COLORED COMBINATIONS
Composition α = (α1, . . . , αm)←→ Young composition diagram
(2,1,3) α=
Given set C = {1, 2, . . . ,P} of colors, consider colored Young diagrams (coloredcompositions, multisequences of colors)
([2,1],[2],[1,1,2])
2 1
2
1 1 2
J=
Fix a field k, the algebra NSymP over k has:linear basis – colored compositionsmuliplication – juxtaposition of such compositions: I.J = I t Junity – empty composition
2 1
2
1 1 2 2
2
2
1
1
2
2
2
2
2 2
1
1 1
1
1=
ADAM DOLIWA (UWM PL) KP MAPS AND CF LANGUAGES 11-17.11.2018 17 / 21
THE HOPF ALGEBRA OF COLORED COMPOSITIONS
PROPOSITION
The algebra NSymP can be equipped with the compatible coproduct defined ongenerators (one-line diagrams) by the left–right splitting, and then extended byhomomorphism
1 1 2∆ ( ) = O 1 1 2 1 1 2+ + 1 1 2 + 1 1 2 O
and with the counit
ε(J ) =
{0 J 6= ∅1 J = ∅
The antipode map in this case (graded and connected bialgebra) is given automatically
REMARKS
In the monochromatic P = 1 case, we obtain the Hopf algebra NSym ofnon-commutative symmetric functions by GELFAND ET AL. [1995]
The above one-line Young composition diagram generators become then thecomplete homogeneous non-commutative symmetric functions
ADAM DOLIWA (UWM PL) KP MAPS AND CF LANGUAGES 11-17.11.2018 18 / 21
THE CONTECT-FREE LANGUAGE OF COLORED COMPOSITIONS AND KPLANGUAGES
REMARKS
Colored compositions←→ (special) words over alphabet A = {a1, b1, . . . , aP , bP}2 1
2
1 1 2
a a b b a b a a a b b b 112211222112
The corrresponding language L∗P is context-free and its characteristic series Σ∗
satisfies equations
Σ = a1Σb1 + · · ·+ aPΣbP
Σ∗ = λ+ ΣΣ∗
here Σ is the characteristic series of the language LP of one-line generators of thealgebra
The support {[ab]α1 , α |= n, n ≥ 0} of the series Z−11 is intersection of L∗P and the
cyclic regular language CP
ADAM DOLIWA (UWM PL) KP MAPS AND CF LANGUAGES 11-17.11.2018 19 / 21
THE GRADED AND SWEEDLER’S DUAL TO NSymP
The Hopf algebra NSymP = ⊕n≥0NSym(n)P is
graded: the weight is the number n we decompose
locally finite: dim NSym(n)P = nP2n−1 <∞
and connected: dim NSym(0)P = 1
therefore by general construction there exists its graded dual (NSymP)gr = QSymP
By (I∗) denote the basis of QSymP dual to (I). The dual coproduct δ to theconcatenation product "·" is
δ(I∗) =∑I=JtK
J ∗ ⊗K∗
Σ∗,Zk BELONG TO THE SWEEDLER DUAL (NSymP)◦
δ(Σ∗) = Σ∗ ⊗ Σ∗, δ(Zk ) = 1⊗ Zk + Zk ⊗ 1
The product in QSymP is the colored version of the quasi-shuffle product AD [2016](natural if you represent elements of QSymP as power series in infinite number ofpartially commuting variables)known from the theory of quasi-symmetric functions (P = 1) by GESSEL [1984]
ADAM DOLIWA (UWM PL) KP MAPS AND CF LANGUAGES 11-17.11.2018 20 / 21
CONCLUSION AND RESEARCH PROBLEMS
The key element in derivation of the Yang-Baxter map from non-commutativediscrete KP hierarchy is characteristic series of certain context-free language
The language can be obtained from generators of the Hopf algebra of generalizednon-commutative symmetric functions (in fact the characteristic series belongs tothe restricted dual of the algebra)
The graded dual to the Hopf algebra NSymP is the Hopf algebra of generalizedquasi-symmetric functions, whose monochromatic version has numerousapplications in algebraic combinatorics
It is well known that Sato approach to the KP hierarchy is deeply embedded in thetheory of symmetric functions — generalize to NSymP and QSymP (or to the Foissyself-dual Hopf algebra of colored trees)
The corresponding theory of Schur functions (the case P = 1 is known)
...
THANK YOU FOR YOUR ATTENTION
ADAM DOLIWA (UWM PL) KP MAPS AND CF LANGUAGES 11-17.11.2018 21 / 21
CONCLUSION AND RESEARCH PROBLEMS
The key element in derivation of the Yang-Baxter map from non-commutativediscrete KP hierarchy is characteristic series of certain context-free language
The language can be obtained from generators of the Hopf algebra of generalizednon-commutative symmetric functions (in fact the characteristic series belongs tothe restricted dual of the algebra)
The graded dual to the Hopf algebra NSymP is the Hopf algebra of generalizedquasi-symmetric functions, whose monochromatic version has numerousapplications in algebraic combinatorics
It is well known that Sato approach to the KP hierarchy is deeply embedded in thetheory of symmetric functions — generalize to NSymP and QSymP (or to the Foissyself-dual Hopf algebra of colored trees)
The corresponding theory of Schur functions (the case P = 1 is known)
...
THANK YOU FOR YOUR ATTENTION
ADAM DOLIWA (UWM PL) KP MAPS AND CF LANGUAGES 11-17.11.2018 21 / 21