tetrahedron equation and generalized quantum...

.

......Tetrahedron equation and generalized quantum groups

Atsuo Kuniba

University of Tokyo

PMNP2015@Gallipoli, 25 June 2015

Atsuo Kuniba (University of Tokyo) Tetrahedron equation and generalized quantum groupsPMNP2015@Gallipoli, 25 June 2015 1 /

1

Key to integrability in 2D

Yang-Baxter equation Reflection equation

R12R13R23 = R23R13R12 R21K2R12K1 = K1R21K2R12

==

1 231 23

12 1

2

R : 2 particle scattering K : Reflection at boundary


1

What about 3D?

Tetrahedron equation (A.B. Zamolodchikov, 1980)

R : F ⊗ F ⊗ F → F ⊗ F ⊗ F (3D R)

R123R145R246R356 = R356R246R145R123

1

12 2

4

4

3

3

5

5

6

6

=

R =

{3 string scattering amplitude in (2+1)D

local Boltzmann weight of the vertex in 3D


1

Status of finding solutions and relevant maths

2D

Infinitely many solutions constructed systematically by representationtheory of the Drinfeld-Jimbo quantum affine algebra Uq(g)(g = affine Kac-Moody algebra).

3D

A few classes of solutions are known.

Systematic framework yet to be developed.

One such approach is by quantized algebra of functions Aq(g) whichis the quantum group corresponding to the dual of Uq(g).(g = finite dimensional simple Lie algebra)


1

Simplest example:

SL2 =

{(t11 t12t21 t22

)| [tij , tkl ] = 0, t11t22 − t12t21 = 1

}.

Aq(sl2) is generated by t11, t12, t21, t22 with the relations

t11t21 = qt21t11, t12t22 = qt22t12, t11t12 = qt12t11, t21t22 = qt22t21,

[t12, t21] = 0, [t11, t22] = (q − q−1)t21t12, t11t22 − qt12t21 = 1.

Hopf algebra with coproduct ∆tij =∑

k tik ⊗ tkj .

Fock representation π1 : Aq(sl2) → End(Fq)

Fq = ⊕m≥0C|m⟩ : q-oscillator Fock space

π1 :

(t11 t12t21 t22

)7−→

(a− k−qk a+

)k|m⟩ = qm|m⟩, a+|m⟩ = |m + 1⟩, a−|m⟩ = (1− q2m)|m − 1⟩.


1

Aq(g) for general g = finite dim. simple Lie algebra

i : a vertex of the Dynkin diagram of g .πi : = irreducible rep. of Aq(g) factoring through Aq(g)→ Aqi (sl2,i )si : = simple reflection in the Weyl group W (g).

.Theorem (Soibelman 1991)..

......

...1 If sii · · · sir ∈W (g) is a reduced word, πi1 ⊗ · · · ⊗ πir is irreducible.

...2 If si1 · · · sir = sj1 · · · sjr are two reduced words, the associated irreps.are equivalent: πi1 ⊗ · · · ⊗ πir ≃ πj1 ⊗ · · · ⊗ πjr .

Corollary: Exists unique (up to overall) intertwiner Φ:

(πi1 ⊗ · · · ⊗ πir ) ◦ Φ = Φ ◦ (πj1 ⊗ · · · ⊗ πjr )


1

Example

Aq(sl3) = ⟨tij⟩3i ,j=1

π1 π2

Fock representations π1 π2t11 t12 t13t21 t22 t23t31 t32 t33

7−→ a− k 0−qk a+ 00 0 1

,

1 0 00 a− k0 −qk a+

W (sl3) = ⟨s1, s2⟩. s2s1s2 = s1s2s1 (Coxeter relation)

=⇒ π2 ⊗ π1 ⊗ π2 ≃ π1 ⊗ π2 ⊗ π1 as representations on (Fq)⊗3

Exists the intertwiner Φ : (Fq)⊗3 → (Fq)

⊗3 such that

(π2 ⊗ π1 ⊗ π2) ◦ Φ = Φ ◦ (π1 ⊗ π2 ⊗ π1).


1

Explicit form

R := ΦP13, P13(x ⊗ y ⊗ z) = z ⊗ y ⊗ x ,

R(|i⟩ ⊗ |j⟩ ⊗ |k⟩) =∑abc

Rabcijk |a⟩ ⊗ |b⟩ ⊗ |c⟩.

Rabcijk = δi+j ,a+bδj+k,b+c

∑λ,µ≥0,λ+µ=b

(−1)λqi(c−j)+(k+1)λ+µ(µ−k)

×[

i , j , c + µ

µ, λ, i − µ, j − λ, c

].

(q)m =m∏j=1

(1− qj),

[i1, . . . , irj1, . . . , js

]=

∏rm=1(q

2)im∏sm=1(q

2)jm


1

.Theorem (Kapranov-Voevodsky 1994)........R satisfies the tetrahedron eq. R123R145R246R356 = R356R246R145R123.

Essence of proof. Consider Aq(sl4) and W (sl4) = ⟨s1, s2, s3⟩.s2s1s2 = s1s2s1, s3s2s3 = s2s3s2, s1s3 = s3s1,

s1s2s3s1s2s1 = s3s2s3s1s2s3 (longest element)

The intertwiner for the last one is constructed in 2 different ways as

123121 Φ456 123121 P34

123212 Φ234 121321 Φ123

132312 P12P45 212321 Φ345

312132 Φ234 213231 P23P56

321232 Φ456 231213 Φ345

321323 P34 232123 Φ123

323123 323123

Equate the 2 sides, substitute Φijk = RijkPik and cancel Pij ’s. □Atsuo Kuniba (University of Tokyo) Tetrahedron equation and generalized quantum groups

PMNP2015@Gallipoli, 25 June 2015 9 /1

Summary so far (type SL case)

Weyl group elements ←→ “Multi-string states”Cubic Coxeter relation ←→ 3D R matrix

Reduced words for longest element ←→ Tetrahedron equation

Remark

3D R = “Quantization” of Miquel’s theorem (1838)(Bazhanov-Sergeev-Mangazeev 2008)

q = 0: set-theoretical sol. to tropical (ultradiscrete) tetrahedron eq.

Recent developments.

......

Type SO, Sp, F4 cases: 3D analogue of reflection equation.

Reduction to 2D: Quantum R’s for generalized quantum groups.

Connection to Poincare-Birkhoff-Witt basis of U+q (g).

Application to multispecies TASEP: Hidden 3D structure

Atsuo Kuniba (University of Tokyo) Tetrahedron equation and generalized quantum groupsPMNP2015@Gallipoli, 25 June 2015 10

/ 1

Aq(sp6) = ⟨tij⟩6i ,j=1: (Reshetikhin-Takhtajan-Faddeev 1990)

π1 π2 π3

Fq Fq Fq2

πk(tij) are given as follows.

π1 :

a− k 0 0 0 0−qk a+ 0 0 0 00 0 1 0 0 00 0 0 1 0 00 0 0 0 a− −k0 0 0 0 qk a+

, π2 :

1 0 0 0 0 00 a− k 0 0 00 −qk a+ 0 0 00 0 0 a− −k 00 0 0 qk a+ 00 0 0 0 0 1

π3 :

1 0 0 0 0 00 1 0 0 0 00 0 A− K 0 00 0 −q2K A+ 0 00 0 0 0 1 00 0 0 0 0 1

, ⟨A±,K⟩ = ⟨a±, k⟩|q→q2 .


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W (sp6) = ⟨s1, s2, s3⟩

s1s3 = s3s1, s1s2s1 = s2s1s2, s2s3s2s3 = s3s2s3s2.

Write simply as πi1,...,ir := πi1 ⊗ · · · ⊗ πir . Then,

Equivalence Intertwiner

π13 ≃ π31, P12(x ⊗ y) = y ⊗ x , (trivial)

π121 ≃ π212, Φ = RP13 (same as SL case),

π2323 ≃ π3232, Ψ = KP14P23 (new).

K ∈ End(Fq2 ⊗ Fq ⊗ Fq2 ⊗ Fq), R ∈ End((Fq)⊗3).


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Explicit form

K (|a⟩ ⊗ |i⟩ ⊗ |b⟩ ⊗ |j⟩) =∑

c,m,d ,n

K cmdna i b j |c⟩ ⊗ |m⟩ ⊗ |d⟩ ⊗ |n⟩.

K c,m,0,na, i , 0, j =

∑λ≥0

(−1)m+λ (q4)c+λ

(q4)cqϕ2

[i , j

λ, j − λ,m − λ, i −m + λ

],

ϕ2 = (a+ c+1)(m+j−2λ)+m−j .

K cmdna i b j =

(q4)a(q4)c

∑α,β,γ≥0

(−1)α+γ

(q4)d−βqϕ1K a,i+b−α−β−γ,0,j+b−α−β−γ

c,m+d−α−β−γ,0,n+d−α−β−γ

×[

b, d − β, i + b − α− β, j + b − α− β

α, β, γ,m − α, n − α, b − α− β, d − β − γ

],

ϕ1 = α(α+2d−2β−1)+(2β−d)(m+n+d)+γ(γ−1)−b(i+j+b).


/ 1

.Theorem (K-Okado 2012)..

......

R and K satisfy the 3D reflection equation (Isaev-Kulish 1997):

R489K3579R269R258K1678K1234R654 = R654K1234K1678R258R269K3579R489.

Two sides come from the 2 ways of constructing the intertwiners for

π123212323 ≃ π323212321 as Aq(sp6)−modules

which correspond to the two reduced words of the longest element

s1s2s3s2s1s2s3s2s3 = s3s2s3s2s1s2s3s2s1 ∈W (sp6).


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Physical and geometric interpretation of the 3D reflection eq.

R489K3579R269R258K1678K1234R654 = R654K1234K1678R258R269K3579R489.

is a “factorization” of 3 string scattering with boundary reflections.

R : Scattering amplitude of 3 strings.K : Reflection amplitude with boundary freedom signified by spaces 1, 3, 7.

1

23 4

56 7

8

9

1

23

456

78

9

=


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F4 case

1 2 3 4

Fq Fq Fq2 Fq2

R : 121 = 212 K : 2323 = 3232 S = R|q→q2 : 434 = 343

π434234232123423123412321 ≃ πreverse order

corresponding to the longest element of W (F4) (length 24) leads to theF4-analogue of the tetrahedron equation:.

......

S14,15,16S9,11,16K16,10,8,7K9,13,15,17S4,5,16R7,12,17S1,2,16R6,10,17S9,14,18K1,3,5,17

× S11,15,18K18,12,8,6S1,4,18S1,8,15R7,13,19K1,6,11,19K4,12,15,19R3,10,19S4,8,11K1,7,14,20

× S2,5,18R6,13,20R3,12,20S1,9,21K2,10,15,20S4,14,21K21,13,8,3S2,11,21S2,8,14R6,7,22

×K2,3,4,22S5,15,21K11,13,14,22R10,12,22K2,6,9,23R3,7,23R19,20,22K16,17,18,22R10,13,23

×K5,12,14,23R3,6,24K16,19,21,23K4,7,9,24R17,20,23K5,10,11,24R12,13,24R17,19,24

×K18,20,21,24S5,8,9R22,23,24 = product in reverse order.


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Now we proceed to the last topic: 2D Reduction

.

......

Tetrahedron equation =⇒ Yang-Baxter equation

R124R135R236R456 = R456R236R135R124

=⇒ R12R13R23 = R23R13R12

Contents

3D L-operator : RLLL = LLLR

Mixed n-product of R and L =⇒ 2n-solutions to YBE

Generalized quantum groups UA(ϵ1, . . . , ϵn),UB(ϵ1, . . . , ϵn) (ϵi = 0, 1)

Main theorem


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3D L-operator: q-oscillator valued 6-vertex model

V = Cv0 ⊕ Cv1, F = Fq, L = (Lγ,δα,β) ∈ End(V ⊗ V ⊗ F )

L(vα ⊗ vβ ⊗ |m⟩) =∑γ,δ

vγ ⊗ vδ ⊗ Lγ,δα,β |m⟩, Lγ,δα,β ∈ End(F )

L0,00,0 = L1,11,1 = 1, L0,10,1 = −qk, L1,01,0 = k, L0,11,0 = a−, L1,00,1 = a+.

L124L135L236R456 = R456L236L135L124 (Bazhanov-Sergeev 2006)

=

6

5

2 14

34

32

1

5

6


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n-layer tetrahedron equation

Write RRRR = RRRR and LLLR = RLLL as(M

(ϵ)αβ4M

(ϵ)αγ5M

(ϵ)βγ6

)R456 = R456

(M

(ϵ)βγ6M

(ϵ)αγ5M

(ϵ)αβ4

), M(0) = R, M(1) = L

M(ϵ)αβ4 ∈ End(

αW (ϵ) ⊗

β

W (ϵ) ⊗4F ), etc, W (0) = F , W (1) = V

∏1≤i≤n

(M

(ϵi )αiβi4

M(ϵi )αiγi5

M(ϵi )βiγi6

)R456 = R456

∏1≤i≤n

(M

(ϵi )βiγi6

M(ϵi )αiγi5

M(ϵi )αiβi4

)

=

5

βn

β1

β2

α1

6

4

γn

αn

γ2

α2

γ1 α1

β1γ1

β2α2

γ2

αn

βnγn

5

4

6Atsuo Kuniba (University of Tokyo) Tetrahedron equation and generalized quantum groups

PMNP2015@Gallipoli, 25 June 2015 19/ 1

2D reduction∏i

(M

(ϵi )αiβi4

M(ϵi )αiγi5

M(ϵi )βiγi6

)R456 = R456

∏i

(M

(ϵi )βiγi6

M(ϵi )αiγi5

M(ϵi )αiβi4

)· · · (♯)

4F ⊗

5F ⊗

6F can be eliminated in two ways:

(A) Tr456(xh4(xy)h5yh6 ( ♯ )

), (B) 456⟨χ|xh4(xy)h5yh6 ( ♯ )|χ⟩456

where [xh4(xy)h5yh6 ,R456] = 0, 456⟨χ|R456 = 456⟨χ|, R456|χ⟩456 = |χ⟩456

h|m⟩ = m|m⟩, |χ⟩456 = |χ⟩4 ⊗ |χ⟩5 ⊗ |χ⟩6, |χ⟩ =∑m≥0

|m⟩(q)m

.

Both lead to YBE: Sα,β(x)Sα,γ(xy)Sβ,γ(y) = Sβ,γ(y)Sα,γ(xy)Sα,β(x) for

(A) Sα,β(z) = Tr3(zh3M

(ϵ1)α1β13

· · ·M(ϵn)αnβn3

),

(B) Sα,β(z) = 3⟨χ|zh3M(ϵ1)α1β13

· · ·M(ϵn)αnβn3

|χ⟩3.

Sα,β(z) ∈ End(W ⊗W ), W = W (ϵ1) ⊗ · · · ⊗W (ϵn).


/ 1

Matrix elements of S(z)

S(z)(|i⟩ ⊗ |j⟩) =∑a,b

Sa,bi,j (z)|a⟩ ⊗ |b⟩,

|a⟩ = |a1, . . . , an⟩ ∈W (ϵ1) ⊗ · · · ⊗W (ϵn), etc.

Matrix element Sa,bi,j (z) is depicted as

)

(A) Trace reduction

b1

6i1

a1q

j1

i2

b2

6

j2

a2q......

.in

anq

bn

6

jn

)

(B) Boundary vector reduction

⟨χ|

b1

6i1

a1q

j1

i2

b2

6

j2

a2q......

.in

anq

bn

6

jn

|χ⟩


/ 1

Problem:

Find a characterization of S(z) obtained by (A) and (B) in the frameworkof the quantum group theory for each choice of (ϵ1, . . . , ϵn) ∈ {0, 1}n.

Example:

(ϵ1, . . . , ϵ5) = (01101).

(A) S(z) = Tr(RLLRL), (B) S(z) = ⟨χ|RLLRL|χ⟩,S(z) ∈ End(W ⊗W ), W = F ⊗ V ⊗ V ⊗ F ⊗ V

.Result..

......

They are quantum R-matrices for some specific representations ofgeneralized quantum groups UA = UA(ϵ1, . . . , ϵn) and UB = UB(ϵ1, . . . , ϵn)defined in the sequel.


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Def. UA(ϵ1, . . . , ϵn), UB(ϵ1, . . . , ϵn) (ϵi = 0, 1)

UA and UB are Hopf algebras over C(q12 ) with generators

ei , fi , k±1i (0 ≤ i ≤ n) and relations (n = n − 1 for UA, n = n for UB)

kik−1i = k−1

i ki = 1, [ki , kj ] = 0,

kiej = Di ,jejki , ki fj = D−1i ,j fjki , [ei , fj ] = δi ,j

ki − k−1i

ri − r−1i

.

p = iq−12 , qi = q ( ϵi = 0), qi = −q−1 ( ϵi = 1),

ri = q for UA, ri =

{p i = 0, n,

p2 0 < i < nfor UB ,

Di ,j =∏

k∈⟨i⟩∩⟨j⟩

qk2δi,j−1, ⟨i⟩ =

{{i , i + 1} for UA,{i , i + 1} ∩ [1, n] for UB .

∆k±1i = k±1

i ⊗ k±1i , ∆ei = 1⊗ ei + ei ⊗ ki , ∆fi = fi ⊗ 1 + k−1

i ⊗ fi


/ 1

Special cases: (up to Serre-type relations)

∀ϵi = 0, 1 cases = quantized Kac-Moody algebras of affine type

UA(0, . . . , 0) = Uq(A(1)n−1), UA(1, . . . , 1) = U−q−1(A

(1)n−1),

UB(0, . . . , 0) = Uq(D(2)n+1), UB(1, . . . , 1) = U−q−1(D

(2)n+1).

UA(κ︷︸︸︷

0, . . . , 0,

κ′︷︸︸︷1, . . . , 1), UB(

κ︷︸︸︷0, . . . , 0,

κ′︷︸︸︷1, . . . , 1)

= affinization of quantum superalgebras of type A and B.

In general, UA and UB are examples of generalized quantum groupsintroduced and being developed by Heckenberger (2010),Andruskiewitsch-Schneider (2010), Angiono-Yamane (2015),Azam-Yamane-Yousofzadeh, etc.


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Relevant irreducible representation πx (UB case)

|m⟩ = |m1, . . . ,mn⟩ ∈W = W (ϵ1)⊗ · · · ⊗W (ϵn) (W (0) = F , W (1) = V )

ei = (0, ...,i1, ..., 0), [m] = (qm − q−m)/(q − q−1)

πx : UB(ϵ1, . . . , ϵn)→ End(W ) is defined by

e0|m⟩ = x |m+ e1⟩, en|m⟩ = [mn]|m− en⟩,f0|m⟩ = x−1[m1]|m− e1⟩, fn|m⟩ = |m+ en⟩,k0|m⟩ = p−1(q1)

m1 |m⟩, kn|m⟩ = p(qn)−mn |m⟩,

ei |m⟩ = [mi ]|m− ei + ei+1⟩ (0 < i < n),

fi |m⟩ = [mi+1]|m+ ei − ei+1⟩ (0 < i < n),

ki |m⟩ = (qi )−mi (qi+1)

mi+1 |m⟩ (0 < i < n).

∀ϵi = 0 case: q-oscillator representation of Uq(D(2)n+1)

∀ϵi = 1 case: spin representation of U−q−1(D(2)n+1)


/ 1

Quantum R matrix

R(z) ∈ End(W ⊗W ) is characterized up to an overall scalar by

[PR(z),∆x ,y (g)] = 0 ∀g ∈ UB ,where ∆x ,y := (πx ⊗ πy )∆, z = x/y , P(u ⊗ v) = v ⊗ u.

.Theorem (K-Okado-Sergeev 2015)..

......

S(z)’s obtained by (A) trace reduction and (B) boundary vector reductionare the quantum R matrices of UA and UB , respectively.


/ 1

Ending remarks

More is known for homogeneous case ϵ1 = · · · = ϵn = 0, 1.

∃ Two boundary vectors |χ1⟩ =∑

m|m⟩(q)m

, |χ2⟩ =∑

m|2m⟩(q4)m

⇐⇒ End shape of relevant Dynkin diagrams

0

⟨χ1|R · · ·R

Uq(D(2)n+1)

|χ1⟩

n 0

⟨χ1|R · · ·R

Uq(A(2)2n )

|χ2⟩

n 0

⟨χ2|R · · ·R

Uq(C(1)n )

|χ2⟩

n

Quantum R matrix for Uq(A(1)n−1) = Tr(LL · · · L)

=⇒ Matrix product formula for steady state probabilityin 1D Totally Asymmetric Simple Exclusion Process (TASEP)(K-Maruyama-Okado, arXiv.1506.04490.)


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A.K. and S. Sergeev, Tetrahedron Equation and quantum R matrices for

spin representations of B(1)n ,D

(1)n and D

(2)n+1. Commun. Math. Phys. 324

(2013) 695–713.

A.K. and M. Okado, Tetrahedron and 3D reflection equations fromquantized algebra of functions. J. Phys. A: 45 (2012) 465206.

Tetrahedron equation and quantum R matrices for

q-oscillator representations of Uq(A(2)2n ),Uq(C

(1)n ) and Uq(D

(2)n+1). Commun.

Math. Phys. 334 (2015) 1219–1244.

A.K., M. Okado and Y. Yamada, A common structure in PBW bases of thenilpotent subalgebra of Uq(g) and quantized algebra of functions. SIGMA 9(2013), 049, 23

A.K., M. Okado and S. Sergeev, Tetrahedron equation and quantum R

matrices for modular double of Uq(D(2)n+1),Uq(A

(2)2n ) and Uq(C

(1)n ). Lett.

Math. Phys. 105 (2015) 447–461.

, Tetrahedron equation and generalizedquantum groups, J. Phys. A to appear.


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