PENTAGON, TETRAHEDRON AND YANG–BAXTER MAPS INNON-COMMUTING VARIABLES, AND THEIR GEOMETRIC ORIGIN

Adam Doliwadoliwa@matman.uwm.edu.pl

University of Warmia and Mazury, Olsztyn, Poland

GEOMETRIC AND ALGEBRAIC ASPECTS OF INTEGRABILITYLONDON MATHEMATICAL SOCIETY EPSRC DURHAM SYMPOSIUM

25TH JULY – 4TH AUGUST 2016

ADAM DOLIWA (UWM PL) PENTAGON, TETRAHEDRON & YANG-BAXTER MAPS 25.07–4.08.2016 1 / 24

OUTLINE

1 GEOMETRIC PENTAGON AND TETRAHEDRON MAPS (WITH R. KASHAEV)The normalization mapThe Veblen mapGeometric tetrahedron map

2 4D CONSISTENT SYSTEMS AND ZAMOLODCHIKOV’S CONDITIONDesargues maps and non-commutative KP systemsMultidimensional discrete conjugate nets

3 YANG–BAXTER MAPSPeriodic reduction of the non-commutative KP mapCentral-product reduction

ADAM DOLIWA (UWM PL) PENTAGON, TETRAHEDRON & YANG-BAXTER MAPS 25.07–4.08.2016 2 / 24

OUTLINE

1 GEOMETRIC PENTAGON AND TETRAHEDRON MAPS (WITH R. KASHAEV)The normalization mapThe Veblen mapGeometric tetrahedron map

2 4D CONSISTENT SYSTEMS AND ZAMOLODCHIKOV’S CONDITIONDesargues maps and non-commutative KP systemsMultidimensional discrete conjugate nets

3 YANG–BAXTER MAPSPeriodic reduction of the non-commutative KP mapCentral-product reduction

ADAM DOLIWA (UWM PL) PENTAGON, TETRAHEDRON & YANG-BAXTER MAPS 25.07–4.08.2016 3 / 24

PENTAGON MAPS

X – a set, a map S : X 2 → X 2 satisfying in X 3 the relation

S12 ◦ S13 ◦ S23 = S23 ◦ S12

is called a pentagon map [S. Zakrzewski 1992], [Semenov-Tian-Shanskii 1992]

PENTAGON PROPERTY OF THE NORMALIZATION MAP

The following birational map N : (x1, x2) 99K (x ′1, x′2)

x ′1 = (x2 + x1 − x1x2)−1 x1, x ′2 = x2 + x1 − x1x2,

satisfies the pentagonal condition

COROLLARY

The inverse of N is given by

x1 = x ′2x ′1, x2 = (1− x ′2x ′1)−1x ′2(1− x ′1),

and satisfies the reversed pentagonal condition

ADAM DOLIWA (UWM PL) PENTAGON, TETRAHEDRON & YANG-BAXTER MAPS 25.07–4.08.2016 4 / 24

GEOMETRY OF THE NORMALIZATION MAP [AD, Sergeev 2014]

Given four collinear points A, B, C and D, consider two pairs of linear relationsbetween their non-homogeneous coordinates

A

B

C

D

1

21 2A

D

C

B

−φ B B

A

C

D

) x1Cφ(=Bφ−Aφ

φA = φCx1 + φB(1− x1)φD = φCx2 + φA(1− x2)

and φA = φDx ′1 + φB(1− x ′1)φD = φCx ′2 + φA(1− x ′2).

The normalization map is a consequence of that change of basis

ADAM DOLIWA (UWM PL) PENTAGON, TETRAHEDRON & YANG-BAXTER MAPS 25.07–4.08.2016 5 / 24

COMBINATORIAL DESCRIPTION OF THE PENTAGONAL CONDITION

S23 12= S12 S

13 S23S

1

2

3

1

2

3

1

2 3

1

2

3

1

2

3

12

12

13

23

23

ADAM DOLIWA (UWM PL) PENTAGON, TETRAHEDRON & YANG-BAXTER MAPS 25.07–4.08.2016 6 / 24

LINEAR PROBLEM FOR THE VEBLEN MAP [AD, Sergeev 2014]

φAC = φABx1 + φAD(1− x1)φBC = φACx2 + φCD(1− x2)

and φBC = φAB x1 + φBD(1− x1)φBD = φAD x2 + φCD(1− x2).

The Veblen (62, 43) configuration

V

AC

A

C

BC

AC

A

C

CD

BC

AD

AB

CD

AB

AD

D

B

D

B

BD BD

A

B

C

D

A

B

C

D

ADAM DOLIWA (UWM PL) PENTAGON, TETRAHEDRON & YANG-BAXTER MAPS 25.07–4.08.2016 7 / 24

(AFFINE) VEBLEN MAP

PENTAGON PROPERTY OF THE VEBLEN MAP

The birational map V : (x , y) 99K (x , y), where

x1 = x1x2, x2 = (1− x1)x2(1− x1x2)−1,

satisfies the reversed pentagonal condition

V23 ◦ V13 ◦ V12 = V12 ◦ V23

Notice that V op = N−1

COROLLARY

The inverse of V is given by

x1 = x1(x1 + x2 − x2x1)−1, x2 = x1 + x2 − x2x1,

and satisfies the pentagonal condition

ADAM DOLIWA (UWM PL) PENTAGON, TETRAHEDRON & YANG-BAXTER MAPS 25.07–4.08.2016 8 / 24

THE VEBLEN MAP AND DESARGUES (103) CONFIGURATION

BDE

ABE

ADE

BCE

ABC

ACE

BCD

CDE

ABD

ACD

BC

BE

DE

AD

AE CD

AC

BD

AB

CE

Pentagonal property of the Veblen map is a consequence of the Desargues theorem• lines are labeled by two-element subsets out of five-letter set• points are labeled by three-element subsets• contains five Veblen configurations labeled by subsets containing a fixed letter

ADAM DOLIWA (UWM PL) PENTAGON, TETRAHEDRON & YANG-BAXTER MAPS 25.07–4.08.2016 9 / 24

TETRAHEDRON MAPS

A map R : X 3 → X 3, which satisfies in X 6 the relation

R123 ◦ R145 ◦ R246 ◦ R356 = R356 ◦ R246 ◦ R145 ◦ R123,

proposed on the quantum level by [Zamolodchikov 1981], is called tetrahedron map

The birational map R = P23 ◦ V12 ◦ N13, : (x1, x2, x3) 99K (x1, x2, x3), where

x1 = [x3 + x1(1− x3)]−1 x1x2, x2 = x3 + x1(1− x3),

x3 = 1 + (x2 − 1) [(1− x1)x3 + x1(1− x2)]−1 (x3 + x1(1− x3)) ,

satisfies the tetrahedron condition

The inverse map R−1 : (x1, x2, x3) 99K (x1, x2, x3) , given explicitly by

x1 = x2x1 [x3 + (1− x3)x1]−1 , x2 = x3 + (1− x3)x1,

x3 = 1 + (x3 + (1− x3)x1) [x3(1− x1) + (1− x2)x1]−1 (x2 − 1),

satisfies the tetrahedron condition as well

ADAM DOLIWA (UWM PL) PENTAGON, TETRAHEDRON & YANG-BAXTER MAPS 25.07–4.08.2016 10 / 24

GEOMETRY OF THE TEN-TERM RELATION FOR THE NORMALIZATION ANDVEBLEN MAPS

THEOREM [Kashaev, Sergeev 1998]

Given a solution N of the functional pentagon equation, and given a solution V of thereversed functional pentagon equation on the same set X , then the mapR = P23 ◦ V12 ◦ N13 satisfies the Zamolodchikov tetrahedron equation, provided

V13 ◦ N12 ◦ V14 ◦ N34 ◦ V24 = N34 ◦ V24 ◦ N14 ◦ V13 ◦ N12.

A

B

AB AC AD

E

DC

BC

BE

DE

AE

CD

BD

Start from seven points (black circles) of the star configuration (102, 54) AND FOUR

CORRESPONDING LINEAR RELATIONS there are two distinct ways to complete theconfiguration using the normalization and Veblen flips

ADAM DOLIWA (UWM PL) PENTAGON, TETRAHEDRON & YANG-BAXTER MAPS 25.07–4.08.2016 11 / 24

OUTLINE

1 GEOMETRIC PENTAGON AND TETRAHEDRON MAPS (WITH R. KASHAEV)The normalization mapThe Veblen mapGeometric tetrahedron map

2 4D CONSISTENT SYSTEMS AND ZAMOLODCHIKOV’S CONDITIONDesargues maps and non-commutative KP systemsMultidimensional discrete conjugate nets

3 YANG–BAXTER MAPSPeriodic reduction of the non-commutative KP mapCentral-product reduction

ADAM DOLIWA (UWM PL) PENTAGON, TETRAHEDRON & YANG-BAXTER MAPS 25.07–4.08.2016 12 / 24

DESARGUES MAPS AND NON-COMMUTATIVE DISCRETE MKP SYSTEM

DESARGUES MAPS [AD 2010]

Maps φ : ZN → PM (D), such that the points φ(n), φ(i)(n) and φ(j)(n) are collinear, for alln ∈ ZN , i 6= j ; here D is a division ring

NOTATION: φ(i)(n1, . . . , ni , . . . , nN) = φ(n1, . . . , ni + 1, . . . , nN)

In non-homogeneous (affine) coordinates we have Φ : ZN → DM

(Φ(j) −Φ) = (Φ(i) −Φ)Bij ,

• the first part of the compatibility condition gives

BijBjk = Bij ,

which allows to introduce a potential σ : ZN → D∗ such that

Bij = σ(i)σ−1(j) ;

• the second part of the compatibility condition takes then the form

(σ−1(i) − σ

−1(j) )σ(ij) + (σ−1

(j) − σ−1(k) )σ(jk) + (σ−1

(k) − σ−1(i) )σ(ki) = 0

known as the non-commutative discrete mKP system [Nijhoff, Capel 1990]ADAM DOLIWA (UWM PL) PENTAGON, TETRAHEDRON & YANG-BAXTER MAPS 25.07–4.08.2016 13 / 24

THE LINEAR PROBLEM FOR THE TETRAHEDRON MAP

Φ(2) = Φx1 + Φ(1)(1− x1), Φ(23) = Φ(2)x2 + Φ(12)(1− x2), Φ(3) = Φx3 + Φ(2)(1− x3)

13

Φ

Φ(123)

Φ(1)

Φ (12)

Φ

Φ(123)

Φ(1)

Φ (12)

R

Φ(3)

Φ(23)Φ(23)

Φ(3)

1 3~~

2

Φ(13)

~2

Φ(2)

After the cube flip we arrive to three new linear relations

Φ(23) = Φ(3)x1+Φ(13)(1−x1), Φ(3) = Φx2+Φ(1)(1−x2), Φ(13) = Φ(1)x3+Φ(12)(1−x3)

Φ

Φ Φ

Φ

Φ

(2)

Φ (1)

Φ(3)

(12)

(23)

(13)

The relation between the Veblen configuration (the Menelaus theorem) and thediscrete Schwarzian KP equation was known to [Konopelchenko, Schief 2002]

ADAM DOLIWA (UWM PL) PENTAGON, TETRAHEDRON & YANG-BAXTER MAPS 25.07–4.08.2016 14 / 24

4D CUBE (TESSERACT) VISUALIZATION OF ZAMOLODCHIKOV’SCONDITION

6

2

15

3

41

5

4

6

2

3

R246

R356

145

123R

35

1

2

4

6

6

5

4

13

26

5

4

2

1 3

246R

1

2

3

4

6

5

24

1

5

3

6

2

4

6

15

3

R123

R356

R

R145

R123

R145

R246

R356

=R356

R246

R145

R123

13

4

2

Φ

Φ (1234)

ADAM DOLIWA (UWM PL) PENTAGON, TETRAHEDRON & YANG-BAXTER MAPS 25.07–4.08.2016 15 / 24

MULTIDIMENSIONAL CONSISTENCY OF DISCRETE CONJUGATE NETS ANDDISCRETE DARBOUX EQUATIONS [AD, Santini 1997]

Qij(k) = Qij + Qik(j)Qkj , i, j, k disctinct

[Bogdanov, Konopelchenko 1995]The corresponding solution of the functional tetrahedron equationwas constructed by [Bazhanov, Sergeev 2006]

ADAM DOLIWA (UWM PL) PENTAGON, TETRAHEDRON & YANG-BAXTER MAPS 25.07–4.08.2016 16 / 24

FURTHER COMMENTS ON DESARGUES MAPS AND MULTIDIMENSIONALQUADRILATERAL LATTICES

Desargues maps naturally are defined on the root lattice Q(AN) [AD 2011]theory of K dimensional discrete conjugate nets is equivalent to theoryof 2K − 1 dimensional Darboux maps [AD 2010]the discrete BKP [Miwa 1982] , and the discrete CKP [Kashaev 1996] equationsin K dimensions are obtained as reductions of the discrete (A)KP [Hirota 1981]equations in 2K − 1 dimensions [AD 2013]

ABC

ABD

ABE

ABFAB

ACD

AC

ADE

AEF

ADF

ACF

AF

AD

ACE

AE

BF

BE

BEF

BDE

BCE

BC

BCF

CF

BCD

BD

BDF

DEF

DE

CDF

DF

CEF

CE

CD CDE

EF

The (203, 154) configuration as "linear" construction of the quadrilateral lattice

ADAM DOLIWA (UWM PL) PENTAGON, TETRAHEDRON & YANG-BAXTER MAPS 25.07–4.08.2016 17 / 24

DESARGUES MAPS IN THE HIROTA GAUGE

In homogeneous coordinates of the projective space, and in a suitable gauge

Φ(i) −Φ(j) = ΦUij , 1 ≤ i 6= j,

whose compatibility is

Uij + Uji = 0, Uij + Ujk + Uki = 0, UkiUkj(i) = UkjUki(j) i, j, k distinct

[Nimmo 2006]When D is commutative then the functions Uij can be parametrized in terms of a singlepotential τ : ZN → D

Uij =ττ(ij)τ(i)τ(j)

, 1 ≤ i < j ≤ N

and the nonlinear system reads [Hirota 1981], [Miwa 1982]

τ(i)τ(jk) − τ(j)τ(ik) + τ(k)τ(ij) = 0, 1 ≤ i < j < k

ADAM DOLIWA (UWM PL) PENTAGON, TETRAHEDRON & YANG-BAXTER MAPS 25.07–4.08.2016 18 / 24

OUTLINE

1 GEOMETRIC PENTAGON AND TETRAHEDRON MAPS (WITH R. KASHAEV)The normalization mapThe Veblen mapGeometric tetrahedron map

2 4D CONSISTENT SYSTEMS AND ZAMOLODCHIKOV’S CONDITIONDesargues maps and non-commutative KP systemsMultidimensional discrete conjugate nets

3 YANG–BAXTER MAPSPeriodic reduction of the non-commutative KP mapCentral-product reduction

ADAM DOLIWA (UWM PL) PENTAGON, TETRAHEDRON & YANG-BAXTER MAPS 25.07–4.08.2016 19 / 24

THE DISCRETE NON-COMMUTATIVE KP HIERARCHY

Let us distinguish the last variable k = nN , denote also

n = (n1, . . . , nN−1), Φ(n, k) = Ψk (n), UN,i (n, k) = ui,k (n)

which allows the rewrite a part (that with the distinguished variable) of the linearproblem in the form [Kajiwara, Noumi, Yamada 2002]

Ψk+1 −Ψk(i) = Ψk ui,k , i = 1, . . . ,N − 1

uj,k ui,k(j) = ui,k uj,k(i), ui,k(j) + uj,k+1 = uj,k(i) + ui,k+1

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)

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 multidimensionaly consistent u i = (ui,k )

k ∈ Zk ∈ ZP , ui,k+P = ui,k

ADAM DOLIWA (UWM PL) PENTAGON, TETRAHEDRON & YANG-BAXTER MAPS 25.07–4.08.2016 20 / 24

FROM KP MAP TO YANG-BAXTER MAP

A map R : X × X → X ×X satisfying the relation

R12 ◦ R13 ◦ R23 = R23 ◦ R13 ◦ R12, in X × X × X

is called Yang–Baxter map [Drinfeld 1992]

iu

u

R

=x

i(j)~

=yju~ uj(i)=y

=x

=

1

2

3R23

R13

R12

1

2

3

12R

R23

R13

OBSERVATION

The companion map of a three dimensionally consistent map gives rise to Yang–Baxtermap [Adler, Bobenko, Suris 2004]

xk yk = yk xk , yk + xk+1 = xk + yk+1

Problem: Find the companion map of the KP map

ADAM DOLIWA (UWM PL) PENTAGON, TETRAHEDRON & YANG-BAXTER MAPS 25.07–4.08.2016 21 / 24

NON-COMMUTATIVE YANG–BAXTER MAPS

THEOREM

Given non-commuting variables x = (x1, . . . , xP), y = (y1, . . . , yP) define

Xk = xk xk+1 . . . xk+P , Yk = yk yk+1 . . . yk+P

Pk =P−1Xa=0

a−1Yi=0

yk+i

P−1Yi=a+1

xk+i

!, k = 1, . . . ,P,

where subscripts should be read modulo P. If hk is a solution of the Sylvester equation

hkXk + Pk+1 = Yk hk (hk =∞Xj=0

Y−j−1k Pk+1X j

k )

thenR(x , y) = (x , y), xk = hk−1xk h−1

k , yk = h−1k−1yk hk ,

is a Yang–Baxter map

Commutative case [Kajiwara, Noumi, Yamada 2002], [Etingof 2003]

ADAM DOLIWA (UWM PL) PENTAGON, TETRAHEDRON & YANG-BAXTER MAPS 25.07–4.08.2016 22 / 24

THE CENTRAL REDUCTION OF THE YANG–BAXTER MAPS [AD 2014]

Assume that the products α = X1 = x1x2 . . . xP , β = Y1 = y1y2 . . . yP are central, then:(i) Xk and Yk do not depend on k(ii) the Yang–Baxter map R(x , y) = (x , y) simplifies to

xk = Pk xkP−1k+1, yk = P−1

k ykPk+1

(iii) the products α and β are conserved under the map R

In the simplest case P = 2: α = x1x2, β = y1y2 we put x = x1, y = y1 to get aparameter dependent reversible Yang–Baxter map R(α, β) : (x , y) 7→ (x , y)

x =“αx−1 + y

”x“

x + βy−1”−1

,

y =“αx−1 + y

”−1y“

x + βy−1”,

which in the commutative case is equivalent to the FIII map in the list of[Adler, Bobenko, Suris 2004]

ADAM DOLIWA (UWM PL) PENTAGON, TETRAHEDRON & YANG-BAXTER MAPS 25.07–4.08.2016 23 / 24

THANK YOU

A. Doliwa, DESARGUES MAPS AND THE HIROTA-MIWA EQUATION, Proc. R. Soc. A466 (2010) 1177-1200

A. Doliwa, S. M. Sergeev, THE PENTAGON RELATION AND INCIDENCE GEOMETRY,J. Math. Phys. 55 (2014) 063504

A. Doliwa, NON-COMMUTATIVE RATIONAL YANG-BAXTER MAPS, Lett. Math. Phys.104 (2014) 299-309

A. Doliwa, R. M. Kashaev, NON-COMMUTATIVE RATIONAL PENTAGON AND

TETRAHEDRON RELATIONS, AND DESARGUES MAPS, in preparation

ADAM DOLIWA (UWM PL) PENTAGON, TETRAHEDRON & YANG-BAXTER MAPS 25.07–4.08.2016 24 / 24