S I D E XIII K U , F , J N 11–17, 2018side13conference.net/presentations/AdamDoliwa.pdf · KP...

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


OUTLINE





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


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


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


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


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 .


OUTLINE





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)∗


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


THE TRIAD: PROBLEM — TOOLS — SYMMETRY

SYMMETRY

TOOLSPROBLEM


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


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〉


OUTLINE





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=


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


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


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]


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


S I D E XIII K U , F , J N 11–17, 2018side13conference.net/presentations/AdamDoliwa.pdf · KP...

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