Quantum loop algebras and functional relations: Oscillator vs. - - PowerPoint PPT Presentation

Quantum loop algebras and functional relations: Oscillator vs. prefundamental representations

Khazret S. Nirov

UNI WUPPERTAL & INR RAS, MOSCOW

RAQIS’16, Geneva, August 23, 2016

1

Introduction

2

Quantum groups

3

Universal integrability objects

4

Uq(gll+1)- and Uq(L(sll+1))-modules

5

Uq(b+)-submodules

6

Universal functional relations

7

Highest ℓ-weight modules

8

Appendix

Hidden Grassmann structure Boos, Jimbo, Miwa, Smirnov, Takeyama, 2006 – 2016 Quantum group approach to functional relations Bazhanov, Lukyanov, Zamolodchikov, 1994 – 1999 Bazhanov, Hibberd, Khoroshkin, 2001 Group-theoretic / Algebraic approach revisited Boos, G¨

• hmann, Kl¨

umper, NX, Razumov, 2010 – 2016 Prefundamental representations and functional relations Hernandez, Jimbo, 2011 Frenkel, Hernandez, 2014

Quantum groups: V. Drinfeld & M. Jimbo (1985 - 1986)

A is a Hopf algebra with respect to ∆, S, ε A is a Hopf algebra with ∆op = Π ◦ ∆ Π(a ⊗ b) = b ⊗ a, a, b ∈ A The universal R-matrix ∆op(a) = R ∆(a)R−1, R ∈ B+ ⊗ B− ⊂ A ⊗ A (∆ ⊗ id)(R) = R13 R23, (id ⊗ ∆)(R) = R13 R12 The master equation R12 R13 R23 = R23 R13 R12 defined in B+ ⊗ A ⊗ B− ⊂ A ⊗ A ⊗ A

Universal integrability objects

Yang-Baxter equation with ∆(t) = t ⊗ t (R13t1)(R23t2) = (R12)−1(R23t2)(R13t1)R12 (∗) Monodromy operator: ϕ : A → End(V) Mϕ(ζ) = (ϕζ ⊗ id)(R) ∈ End(V) ⊗ A Transfer operator Tϕ(ζ) = (trV ⊗ id)(Mϕ(ζ)(ϕζ(t) ⊗ 1)) = ((trV ◦ϕζ) ⊗ id)(R(t ⊗ 1)) L-operator: ρ : B+ → End(W) Lρ(ζ) = (ρζ ⊗ id)(R) ∈ End(W) ⊗ A Q-operator Qρ(ζ) = (trW ⊗id)(Lρ(ζ)(ρζ(t) ⊗ 1)) = ((trW ◦ρζ) ⊗ id)(R(t ⊗ 1))

First universal functional relations

Directly from the Yang-Baxter equation (tr ◦ϕ1ζ1) ⊗ (tr ◦ϕ2ζ2)(∗) ⇒ Tϕ1(ζ1)Tϕ2(ζ2) = Tϕ2(ζ2)Tϕ1(ζ1) (tr ◦ρζ1) ⊗ (tr ◦ϕζ2)(∗) ⇒ Qρ(ζ1)Tϕ(ζ2) = Tϕ(ζ2)Qρ(ζ1) The rest is more tricky: Qρ1(ζ1)Qρ2(ζ2) = ((trW1⊗W2 ◦(ρ1ζ1 ⊗ ρ2ζ2)) ⊗ id)

• R13t1R23t2

R13t1R23t2 = [(∆ ⊗ id)(R)] [(∆ ⊗ id)(t ⊗ 1)] = (∆ ⊗ id)(R(t ⊗ 1)) Qρ1(ζ1)Qρ2(ζ2) = ((trW1⊗W2 ◦(ρ1ζ1 ⊗∆ ρ2ζ2)) ⊗ id)(R(t ⊗ 1)) Cipher key C Tλ(ζ) = Q1(q2(λ+ρ)1/sζ)Q2(q2(λ+ρ)2/sζ)Q3(q2(λ+ρ)3/sζ) C Tλ(ζ) = Q1(q−2(λ+ρ)1/sζ)Q2(q−2(λ+ρ)2/sζ)Q3(q−2(λ+ρ)3/sζ)

Quantum group Uq(gll+1)

Roots : normal ordering of △+ [ l(l + 1)/2 positive roots ] α1, α1 + α2, α2, α1 + α2 + α3, α2 + α3, α3, . . . . . . α1 + α2 + . . . + αi, α2 + . . . + αi, . . . , αi, . . . . . . α1 + α2 + . . . + αl, α2 + . . . + αl, . . . , αl Equivalent to the total co-lexicographic order ≤ on the set N × N Generators and Cartan subalgebra of gll+1 Ei, Fi, i = 1, . . . , l Ki, i = 1, . . . , l + 1, kl+1 =

l+1

• i=1

CKi Quantum group Uq(gll+1) generated by Ei, Fi, i = 1, . . . , l, qX, X ∈ kl+1 Representations of Uq(gll+1)

Infinite dimensional representation πλ with the highest weight vector vλ Eivλ = 0, i = 1, . . . , l, qXvλ = qλ,Xvλ, X ∈ kl+1, λ ∈ k∗

l+1

Finite dimensional representation πλ arises as a quotient sub-representation from πλ when λi − λi+1 ∈ Z+ for all i = 1, . . . , l

Higher root vectors and q-commutation relations

Root vectors (M. Jimbo, 1986) Ei,i+1 = Ei, i = 1, . . . , l Eij = Ei,j−1 Ej−1,j − q Ej−1,j Ei,j−1, j − i > 1 and Fi,i+1 = Fi, i = 1, . . . , l Fij = Fj−1,j Fi,j−1 − q−1Fi,j−1 Fj−1,j, j − i > 1 Simplest commutations qνKi Emn q−νKi = qν ∑n−1

j=m cijEmn,

qνKi Fmn q−νKi = q−ν ∑n−1

j=m cijFmn

qνHi = qν(Ki−Ki+1), i = 1, . . . , l qνHi Emn q−νHi = qν ∑n−1

j=m aijEmn,

qνHi Fmn q−νHi = q−ν ∑n−1

j=m aijFmn

where cij : (l + 1) × l matrix with cii = 1, ci+1,i = −1, ci,i+1 = 0, i = 1, . . . , l, cij = 0 for |j − i| ≥ 2 αj(Ki) = cij, αj(Ki) − αj(Ki+1) = αj(Hi) = aij, i, j = 1, . . . , l Six different branches for {(ij), (mn)} ∈ N2 × N2 (H. Yamane, 1989) CI : i = m < j < n, CII : m < i < j < n, CIII : i < m < j = n CIV : i < m < j < n, CV : i < j = m < n, CVI : i < j < m < n

Further commutation relations

Eij with Emn Eij Emn = q−1Emn Eij, {(ij), (mn)} ∈ CI ∪ CIII Eij Emn = Emn Eij, {(ij), (mn)} ∈ CII ∪ CVI Eij Emn − q Emn Eij = Ein, {(ij), (mn)} ∈ CV Eij Emn − Emn Eij = −κq Ein Emj, {(ij), (mn)} ∈ CIV Fij with Fmn Fij Fmn = q−1Fmn Fij, {(ij), (mn)} ∈ CI ∪ CIII Fij Fmn = Fmn Fij, {(ij), (mn)} ∈ CII ∪ CVI Fij Fmn − q Fmn Fij = −q Fin, {(ij), (mn)} ∈ CV Fij Fmn − Fmn Fij = −κq Fin Fmj, {(ij), (mn)} ∈ CIV

Further commutation relations

Eij with Fmn [Eij , Fij] = κ−1

q

• q∑

j−1 k=i Hk − q− ∑ j−1 k=i Hk

• [Eij , Fmn] = −q−1Fjn q− ∑

j−1 k=i Hk,

{(ij), (mn)} ∈ CI [Eij , Fmn] = q Eim q− ∑n−1

k=m Hk = q− ∑n−1 k=m HkEim,

{(ij), (mn)} ∈ CIII [Eij , Fmn] = 0, {(ij), (mn)} ∈ CII ∪ CV ∪ CVI [Eij , Fmn] = κq Eim Fjn q− ∑

j−1 k=m Hk,

{(ij), (mn)} ∈ CIV In the last relation [Eim, Fjn] = 0 and q− ∑

j−1 k=m Hk Fjn = q−1Fjn q− ∑ j−1 k=m Hk,

q− ∑

j−1 k=m Hk Eim = q Eim q− ∑ j−1 k=m Hk

Further commutation relations

Emn with Fij [Emn , Fij] = −q q∑

j−1 k=i Hk Ejn = −Ejn q∑ j−1 k=i Hk,

{(ij), (mn)} ∈ CI [Emn , Fij] = q−1q∑n−1

k=m Hk Fim = Fim q∑n−1 k=m Hk,

{(ij), (mn)} ∈ CIII [Emn , Fij] = 0, {(ij), (mn)} ∈ CII ∪ CV ∪ CVI [Emn , Fij] = −κq Fim Ejn q∑

j−1 k=m Hk,

{(ij), (mn)} ∈ CIV In the last relation [Ejn , Fim] = 0 and q∑

j−1 k=m Hk Ejn = q−1Ejn q∑ j−1 k=m Hk,

q∑

j−1 k=m Hk Fim = q Fim q∑ j−1 k=m Hk

The elements qνKi, i = 1, . . . , l + 1, Eij, Fij, i = 1, . . . , l, j = 1, . . . , l + 1, i < j generate a P. B. W. basis of Uq(gll+1) as {Fi1j1 . . . Fiaja qν1K1 . . . qνcKc Em1n1 . . . Embnb | a, b, c ≥ 0} where (i1j1) ≤ . . . ≤ (iaja) and (m1n1) ≤ . . . ≤ (mana)

Uq(gll+1)-module Vλ

Basis Eivλ = 0, i = 1, . . . , l, qXvλ = qλ(X)vλ, X ∈ kl+1, λ ∈ k∗

l+1

• Vλ ∼
• qνKivm, Fivm, Eivm | vm = Fm12

12 Fm13 13 Fm23 23 · · · Fm1,l+1 1,l+1 · · · Fml,l+1 l,l+1 v0

• m =
• m12, m13, m23, . . . , m1,l+1, . . . , ml,l+1

∈ Z⊗(l+1)l/2

+

Acting by qνKi and qνHi on vm qνKi vm = qν(λi+∑i−1

k=1 mki−∑l+1 k=i+1 mik)vm,

i = 1, . . . , l + 1 qνHi vm = qν[λi−λi+1+∑i−1

k=1(mki−mk,i+1)−2mi,i+1−∑l+1 k=i+2(mik−mi+1,k)]vm,

i = 1, . . . , l Acting by Fk,k+1 on vm Fk,k+1 vm = q− ∑k−1

i=1 (mik−mi,k+1) vm+ǫk,k+1

+

k−1

∑

j=1

q− ∑

j−1 i=1(mik−mi,k+1) [mjk]q vm−ǫjk+ǫj,k+1,

k = 1, . . . , l Acting by F1,l+1 on vm F1,l+1 vm = q∑l

i=2 m1i vm+ǫ1,l+1

Uq(gll+1)-module Vλ

Basis Eivλ = 0, i = 1, . . . , l, qXvλ = qλ(X)vλ, X ∈ kl+1, λ ∈ k∗

l+1

• Vλ ∼
• qνKivm, Fivm, Eivm | vm = Fm12

12 Fm13 13 Fm23 23 · · · Fm1,l+1 1,l+1 · · · Fml,l+1 l,l+1 v0

• m =
• m12, m13, m23, . . . , m1,l+1, . . . , ml,l+1

∈ Z⊗(l+1)l/2

+

Acting by Ek,k+1 on vm Ek,k+1 vm = qλk−λk+1−2mk,k+1−∑l+1

s=k+2(mks−mk+1,s)

×

k−1

∑

j=1

q∑k−1

i=j+1(mik−mi,k+1) [mj,k+1]q vm+ǫjk−ǫj,k+1

+ [λk − λk+1 −

l+1

∑

s=k+2

(mks − mk+1,s) − mk,k+1 + 1]q [mk,k+1]q vm−ǫk,k+1 −

l+1

∑

j=k+2

q−λk+λk+1−2+∑l+1

i=j (mki−mk+1,i) [mkj]q vm−ǫkj+ǫk+1,j,

k = 1, . . . , l

Quantum loop algebra Uq(L(sll+1))

Lie algebra L(sll+1): generators, Cartan subalgebra and center ei, fi, hi, i = 0, 1, . . . , l,

• hl+1 =

l

• i=0

Chi, c =

l

∑

i=0

hi Quantum group Uq( L(sll+1)) generated by ei, fi, i = 0, 1, . . . , l, qx, x ∈ hl+1 There is no finite dimensional representation of L(sll+1) with c = 0, hence L(sll+1) = L(sll+1)/Cc Uq( L(sll+1)) has no finite dimensional representation with qνc = 1, hence Uq(L(sll+1)) = Uq( L(sll+1))/qνc − 1ν∈C The monomials

• f m1

γ1 · · · f ma γa qx en1 δ1 · · · enb δb

| mi, ni ≥ 0

• with a normal order of the roots γ1 . . . γa,

δ1 . . . δb, form a Poincar´ e–Birkhoff–Witt basis of the quantum loop algebra Uq(L(sll+1))

Jimbo’s homomorphism and representations of Uq(L(sll+1))

Basic representation

• ϕλ

ζ =

πλ ◦ ε ◦ Γ

ζ

Γ

ζ(qx) = qx,

Γ

ζ(ei) = ζsiei,

Γ

ζ(fi) = ζ−sifi

ε : Uq(L(sll+1)) → Uq(gll+1) ε(qνh0) = qν(Kl+1−K1), ε(qνhi) = qν(Ki−Ki+1) ε(e0) = F1,l+1 qK1+Kl+1, ε(ei) = Ei,i+1 ε(f0) = E1,l+1 q−K1−Kl+1, ε(fi) = Fi,i+1 Uq(L(sll+1))-module relations qνh0 vm = q−ν[λ1−λl+1−∑l

i=2(m1i+mi,l+1)−2m1,l+1] vm

qνhi vm = qν[λi−λi+1+∑i−1

k=1(mki−mk,i+1)−2mi,i+1−∑l+1 k=i+2(mik−mi+1,k)]vm

i = 1, . . . , l

Jimbo’s homomorphism and representations of Uq(L(sll+1))

Uq(L(sll+1))-module relations e0 vm = ζs0 qλ1+λl+1+∑l

i=2 mi,l+1 vm+ǫ1,l+1

ek vm = ζsk qλk−λk+1−2mk,k+1−∑l+1

s=k+2(mks−mk+1,s)

×

k−1

∑

j=1

q∑k−1

i=j+1(mik−mi,k+1) [mj,k+1]q vm+ǫjk−ǫj,k+1

+ ζsk [λk − λk+1 −

l+1

∑

s=k+2

(mks − mk+1,s) − mk,k+1 + 1]q [mk,k+1]q vm−ǫk,k+1 − ζsk

l+1

∑

j=k+2

q−λk+λk+1−2+∑l+1

i=j (mki−mk+1,i) [mkj]q vm−ǫkj+ǫk+1,j

fk vm = ζ−sk q− ∑k−1

i=1 (mik−mi,k+1) vm+ǫk,k+1

+ ζ−sk

k−1

∑

j=1

q− ∑

j−1 i=1(mik−mi,k+1) [mjk]q vm−ǫjk+ǫj,k+1

k = 1, . . . , l

Degenerations of the shifted Uq(b+)-modules

Let ϕλ

ζ be a representation of Uq(L(sll+1)), and ξ ∈

h∗

l+1. Then

• ϕλ

ζ [ξ](ei) =

ϕλ

ζ (ei),

• ϕλ

ζ [ξ](qx) = qξ(x)

ϕλ

ζ (qx)

is a representation of Uq(b+) called a shifted representation Universal integrability objects after the module shift T

ϕλ[ξ](ζ) = T ϕλ(ζ) q∑l

i=0 ξ(hi)(hi+φi)/(l+1)

Specific shifts ξ ∈ h∗

l+1

ξ(h0) = λ1 − λl+1, ξ(hi) = −λi + λi+1, i = 1, . . . , l Transforming the basis vm → wm =

l

∏

k=1

cγk(m)

k

vm ck = qλk−λk+1+1+(2λl+1−l)sk/s, γk(m) =

l+1

∑

j=k+1 k

∑

i=1

mij

Degenerations of the shifted Uq(b+)-modules

We have wm = q∑l

k=1 δk γk(m) vm,

δk = λk − λk+1 + 1 + (2λl+1 − l)sk/s so that q∑l

k=1 δk γk(m±ǫij) = q± ∑ j−1 n=i δn q∑l k=1 δk γk(m)

New module after sending λk+1 − λk to infinity qνh0 wm = qν(∑l

i=2(m1i+mi,l+1)+2m1,l+1) wm

qνhi wm = qν(∑i−1

k=1(mki−mk,i+1)−2mi,i+1−∑l+1 k=i+2(mik−mi+1,k))wm

e0 wm = ζs0 q∑l

i=2 mi,l+1 wm+ǫ1,l+1

ek wm = − ζsk κ−1

q

q∑l+1

s=k+2(mks−mk+1,s)+mk,k+1[mk,k+1]q wm−ǫk,k+1

− ζsk

l+1

∑

j=k+2

q∑l+1

i=j (mki−mk+1,i)−1 [mkj]q wm−ǫkj+ǫk+1,j

The new spectral parameter

• ζ = q(2λl+1−l)/s ζ

Invariant Uq(b+)-subspaces

ρ′′ and W′′ for Uq(b+) qνh0 vm = qν(∑l

i=2(m1i+mi,l+1)+2m1,l+1) vm

qνhk vm = qν(∑k−1

i=1 (mik−mi,k+1)−2mk,k+1−∑l+1 i=k+2(mki−mk+1,i))vm

e0 vm = q∑l

i=2 mi,l+1 vm+ǫ1,l+1

ek vm = −κ−1

q

q∑l+1

s=k+2(mks−mk+1,s)+mk,k+1[mk,k+1]q vm−ǫk,k+1

−

l+1

∑

j=k+2

q∑l+1

i=j (mki−mk+1,i)−1 [mkj]q vm−ǫkj+ǫk+1,j

Introduce l(l − 1)/2-tuple p of nonnegative integers pij p =

• p12, p13, p23, . . . p1j, . . . pj−1,j, . . . p1l, . . . pl−1,l
• such that pj−2,j ≤ pj−1,j for every i = 3, . . . , l, and restricted by the

condition

i

∑

s=1

(−1)i−s psj ≥ 0, 1 ≤ i < j ≤ l

Invariant Uq(b+)-subspaces

The subspaces generated by the vectors vm with the indices subject to m12 ≤ p12 m13 ≤ p13, m13 + m23 ≤ p23 m14 ≤ p14, m14 + m24 ≤ p24, m24 + m34 ≤ p34 . . . m1j ≤ p1j, . . . mj−3,j + mj−2,j ≤ pj−2,j, mj−2,j + mj−1,j ≤ pj−1,j . . . m1,l ≤ p1,l, . . . ml−3,l + ml−2,l ≤ pl−2,l, ml−2,l + ml−1,l ≤ pl−1,l are invariant with respect to the action of the quantum group Uq(b+) Define m0j = 0, then mi−1,j + mij ≤ pij, 1 ≤ i < j ≤ l We denote such a Uq(b+)-submodule by W′′

p and introduce a partial

• rder for the l(l − 1)/2-tuples p by assuming that p′ ≤ p if p′

ij ≤ pij for all

possible i = 1, . . . , l − 1 and j = 2, . . . , l

Invariant Uq(b+)-subspaces

ρ′ and W′ for Uq(b+) qνh0 vm = qν(2m1+∑l

i=2 mi) vm

qνhl vm = q−ν(2ml+∑l−1

i=1 mi) vm

qνhj vm = qν(mj+1−mj)vm, j = 1, . . . , l − 1 e0 vm = q∑l

i=2 mi vm+ǫ1,

el vm = −κ−1

q

qml [ml]q vm−ǫl ej vm = −qmj−mj+1−1 [mj]q vm−ǫj+ǫj+1, j = 1, . . . , l − 1 NB! See the isomorphism W′′

p /

• p′≤p

W′′

p′ ∼

= W′[ξp] ξp(h0) =

l

∑

i=2

p1i, ξp(hl) =

l−1

∑

i=1 i

∑

j=1

(−1)i−j pj,l ξp(hj) =

j−1

∑

i=1 i

∑

s=1

(−1)i−s (psj − ps,j+1) − 2

j

∑

s=1

(−1)j−s ps,j+1 −2

l

∑

i=j+2 j

∑

s=1

(−1)j−s psi +

l

∑

i=j+2

pj+1,i, j = 1, . . . , l − 1

Interpretation in terms of q-oscillators: q = exp ¯ h

Oscq is a unital associative C-algebra with generators b†, b, qνN, ν ∈ C q0 = 1, qν1Nqν2N = q(ν1+ν2)N qνNb†q−νN = qνb†, qνNbq−νN = q−νb b†b = [N]q, bb† = [N + 1]q (b†)k+1qνN, bk+1qνN, qνN | k ∈ Z+, ν ∈ C

• is a basis of Oscq

Representations of Oscq

W+ , χ+ : The relations qνNvm = qνmvm b†vm = vm+1, b vm = [m]qvm−1 where v−1 = 0, endow the free vector space generated by {v0, v1, . . .} with the structure of an Oscq-module W− , χ− : The relations qνNvm = q−ν(m+1)vm b vm = vm+1, b† vm = −[m]qvm−1 where v−1 = 0, endow the free vector space generated by {v0, v1, . . .} with the structure of an Oscq-module

Interpretation in terms of q-oscillators: q = exp ¯ h

Consider the algebra Oscq ⊗ . . . ⊗ Oscq = Osc⊗l

q

and define bi = 1 ⊗ . . . ⊗ b ⊗ . . . ⊗ 1, b†

i = 1 ⊗ . . . ⊗ b† ⊗ . . . ⊗ 1,

qνNi = 1 ⊗ . . . ⊗ qνN ⊗ . . . ⊗ 1, Consider W′ and ρ′ in terms of the q-oscillators qνh0 vm = qν(2N1+∑l

i=2 Ni) vm,

qνhl vm = q−ν(2Nl+∑l−1

i=1 Ni) vm

qνhi vm = qν(Ni+1−Ni) vm, i = 1, . . . , l − 1 e0 vm = b†

1 q∑l

i=2 Ni vm,

el vm = −κ−1

q

bl qNl vm ei vm = −bi b†

i+1 qNi−Ni+1−1 vm,

i = 1, . . . , l − 1 Define a homomorphism ρ : Uq(b+) → Osc⊗l

q

by ρ(qνh0) = qν(2N1+∑l

i=2 Ni),

ρ(qνhl) = q−ν(2Nl+∑l−1

i=1 Ni)

ρ(qνhi) = qν(Ni+1−Ni), i = 1, . . . , l − 1 ρ(e0) = b†

1 q∑l

i=2 Ni,

ρ(el) = −κ−1

q

bl qNl ρ(ei) = −bi b†

i+1 qNi−Ni+1−1,

i = 1, . . . , l − 1

More universal integrability objects

Automorphisms σ and τ σ(A(1)

l

) : αi → αi+1, i = 0, 1, . . . , l, σl+1 = id τ(A(1)

l

) : α0 → α0, αi → αl+1−i, i = 1, . . . , l, τ2 = id And more homomorphisms ρ : Uq(b+) → W, ρζ = ρ ◦ Γ

ζ,

θζ = χ ◦ ρζ ρa = ρ ◦ σ−a, ρa = ρ ◦ τ ◦ σ−a+1 Universal L-operators La(ζ) = (ρaζ ⊗ id)(R), La(ζ) = (ρaζ ⊗ id)(R) Universal Q-operators Qa(ζ) = ((tr ◦θaζ) ⊗ id)(R(t ⊗ 1)), Qa(ζ) = ((tr ◦θaζ) ⊗ id)(R(t ⊗ 1))

Main functional relations

Cipher key: the product representation C Tλ(ζ) = Q1(q2(λ+ρ)1/sζ) · · · Ql+1(q2(λ+ρ)l+1/sζ) C = C1 · · · Cl+1, Ci = q−Di/s(q2Dj/s − q2Dk/s)−1 The element ρ ∈ k∗

l+1 is the half-sum of the positive roots

ρ = (l/2, (l − 2)/2, . . . , −l/2) The quantum B.G.G. resolution connects the traces trλ = ∑

w∈W

(−1)ℓ(w) tr w·λ, Tλ(ζ) = ∑

p∈Sl+1

sgn(p) T p(λ+ρ)−ρ(ζ) and leads to the determinant representation (i, j = 1, . . . , l + 1) C Tλ−ρ(ζ) = det

• Qi(q2λj/sζ)
• ,

C Tλ−ρ(ζ) = det

• Qi(q−2λj/sζ)

Khoroshkin–Tolstoy prescription

The higher root vectors eγ+nδ = [(γ|γ)]−1

q [eγ+(n−1)δ, e′ δ, γ]q,

fγ+nδ = [(γ|γ)]−1

q [f ′ δ, γ, fγ+(n−1)δ]q

e(δ−γ)+nδ = [(γ|γ)]−1

q [e′ δ, γ, e(δ−γ)+(n−1)δ]q

f(δ−γ)+nδ = [(γ|γ)]−1

q [f(δ−γ)+(n−1)δ, f ′ δ, γ]q

e′

nδ, γ = [eγ+(n−1)δ, eδ−γ]q,

f ′

nδ, γ = [fδ−γ, fγ+(n−1)δ]q

where [eα , eβ]q = eα eβ − q−(α,β) eβ eα Preparing for e′

nδ,αi

eδ−α1 = [eα2 , [eα3 , . . . [eαl , eδ−θ]q . . .]q ]q eδ−αi = [eαi+1 , . . . [eαl , [eαi−1 , . . . [eα1 , eδ−θ]q . . .]q ]q . . .]q, i = 2, . . . , l − 1 eδ−αl = [eαl−1 , . . . [eα2 , [eα1 , eδ−θ]q]q . . .]q e′

δ,αi = [eαi , eδ−αi]q,

i = 1, . . . , l

Drinfeld–Jimbo’s versus 2nd Drinfeld’s

Generators ξ±

i, n and χi, n

ξ+

i, n =

• (−1)non

i eαi+nδ

n ≥ 0 (−1)n+1on

i q−hi i

f(δ−αi)−(n+1)δ n < 0 ξ−

i, n =

• (−1)non+1

i

e(δ−αi)+(n−1)δ qhi

i

n > 0 (−1)non

i fαi−nδ

n ≤ 0 χi, n =

• (−1)n+1on

i enδ, αi

n > 0 (−1)n+1on

i f−nδ, αi

n < 0 where oi is either +1 or −1, so that oi = −oj whenever aij < 0 Generators φ±

i, n

φ+

i, n =

• (−1)n+1on

i κq qhi i e′ nδ, αi

n > 0 qhi

i

n = 0 φ−

i, n =

• q−hi

i

n = 0 (−1)non

i κq q−hi i

f ′

−nδ, αi

n < 0 φ+

i (u) = qhi i

• 1 − κq e′

δ, αi(−oiu)

• ,

φ−

i (u−1) = q−hi i

• 1 + κq f ′

δ, αi(−oiu−1)

Highest ℓ-weight Uq(L(g))-modules

Highest ℓ-weight Uq(L(g))-module V (in the category O) with highest ℓ-weight Ψ (I = {1, . . . , l}) ∃ v ∈ V : φ±

i,±n v = Ψ± i,±n v,

Ψ±

i,0 = q± λ, hi ,

i ∈ I, n ∈ Z+ ξ+

i,n v = 0,

i ∈ I, n ∈ Z, V = Uq(L(g)) v Rational ℓ-weights Ψ = {Ψ+

i (u), Ψ− i (u−1)}i∈I

Ψ+

i (u) =

aipiupi + ai, pi−1upi−1 + · · · + ai0 bipiupi + bi, pi−1upi−1 + · · · + bi0 = ∑

n∈Z+

Ψ+

i,n un

Ψ−

i (u−1) =

aipi + ai, pi−1u−1 + · · · + ai0u−pi bipi + bi, pi−1u−1 + · · · + bi0u−pi = ∑

n∈Z+

Ψ−

i,−n u−n

Restricted by aik, bik ∈ C, i ∈ I, 0 ≤ k ≤ pi aipi, ai0, bipi, bi0 = 0, aipi bipi ai0 bi0 = 1 { Rational ℓ-weights } ↔ {[ Simple Uq(L(g))-modules in O ]}

Uq(L(sll+1))-modules Vλ and Vλ

Uq(L(sll+1))-modules out of Uq(gll+1)-modules

• Vλ :

ϕλ = πλ ◦ ε, Vλ : ϕλ = πλ ◦ ε ε : Uq(L(sll+1)) → Uq(gll+1)

• Vλ and Vλ are highest ℓ-weight modules in the category O

Automorphisms σ and τ (σl+1 = id, τ2 = id) σ(qh0) = qh1, σ(qh1) = qh2, . . . σ(qhl) = qh0 σ(e0) = e1, σ(e1) = e2, . . . σ(el) = e0 σ(f0) = f1, σ(f1) = f2, . . . σ(fl) = f0 τ(qh0) = qh0, τ(qhi) = qhl−i+1, i ∈ I τ(e0) = e0, τ(ei) = el−i+1, τ(f0) = f0, τ(fi) = fl−i+1 In general, ϕλ

a =

ϕλ ◦ σ−a and ϕλ

a = ϕλ ◦ σ−a are not highest ℓ-weight

• representations. But

ϕλ = ϕλ ◦ τ and ϕλ = ϕλ ◦ τ give new highest ℓ-weight representations θa and θa give highest ℓ-weight representations

Highest ℓ-weight Uq(b+)-modules

Highest ℓ-weight Uq(b+)-module W (in the category O) with highest ℓ-weight Ψ (I = {1, . . . , l}) ∃ v ∈ W : φ+

i,n v = Ψ+ i,n v,

i ∈ I, n ∈ Z+ ξ+

i,n v = 0,

i ∈ I, n ∈ Z+, W = Uq(b+) v Rational ℓ-weights Ψ = {Ψ+

i (u)}i∈I

Ψ+

i (u) =

aipiupi + ai, pi−1upi−1 + · · · + ai0 biqiuqi + bi, qi−1uqi−1 + · · · + bi0 = ∑

n∈Z+

Ψ+

i,n un

Restricted by air, bis ∈ C, ai0, bi0 = 0 i ∈ I, 0 ≤ r ≤ pi, 0 ≤ s ≤ qi { Rational ℓ-weights } ↔ {[ Simple Uq(b+)-modules in O ]} Prefundamental representations L±

i,a :

Ψ+

i (u) = (1, . . . , 1 i−1

, (1 − au)±1, 1, . . . , 1

l−i

), i ∈ I, a ∈ C× Lξ : Ψ+

i (u) = q ξ, hi ,

i ∈ I, ξ ∈ h∗

l+1

Homomorphisms ρa and ρa & representations θa and θa

We recall ρa = ρ ◦ σ−a, ρa = ρ ◦ τ ◦ σ−a+1, a = 1, . . . , l + 1 and define ρa(qνhi) = qν(Ni−a+1−Ni−a), i = a + 1, . . . , l, . . . , l + a − 1 ρa(qνha−1) = q−ν(2Nl+∑l−1

j=1 Nj),

ρa(qνha) = qν(2N1+∑l

j=2 Nj)

ρa(ei) = −bi−a b†

i−a+1 qNi−a−Ni−a+1−1,

i = a + 1, . . . , l, . . . , l + a − 1 ρa(ea−1) = −κ−1

q

bl qNl, ρa(ea) = b†

1 q∑l

j=2 Nj

Representations θa = (χ− ⊗ · · · ⊗ χ−

• l−a+1

⊗ χ+ ⊗ · · · ⊗ χ+

• a−1

) ◦ ρa, a = 1, . . . , l + 1 Appropriate bases v(a)

m = bm1 1 · · · bml−a+1 l−a+1 b† ml−a+2 l−a+2 · · · b† ml l

v0

Homomorphisms ρa and ρa & representations θa and θa

We recall ρa = ρ ◦ σ−a, ρa = ρ ◦ τ ◦ σ−a+1, a = 1, . . . , l + 1 and define ρa(qνhi) = qν(Na−i−Na−i−1), i = 0, 1, . . . , a − 2 ρa(qνha−1) = qν(2N1+∑l

j=2 Nj),

ρa(qνha) = q−ν(2Nl+∑l−1

j=1 Nj)

ρa(qνhi) = qν(Nl+a−i+1−Nl+a−i), i = a + 1, a + 2, . . . , l ρa(ei) = −ba−i−1 b†

a−i qNa−i−1−Na−i−1,

i = 0, 1, . . . , a − 2 ρa(ea−1) = b†

1 q∑l

j=2 Nj,

ρa(ea) = −κ−1

q

bl qNl ρa(ei) = −bl+a−i b†

l+a−i+1 qNl+a−i−Nl+a−i+1−1,

i = a + 1, a + 2, . . . , l Representations θa = (χ− ⊗ · · · ⊗ χ−

• a−1

⊗ χ+ ⊗ · · · ⊗ χ+

• l−a+1

) ◦ ρa, a = 1, . . . , l + 1 Appropriate bases ¯ v(a)

m = bm1 1 · · · bma−1 a−1 b† ma a

· · · b† ml

l

v0

Uq(b+) highest ℓ-weights: a = 1

The ℓ-weights Ψ+

1, m, 1(u) =

q−2m1−∑l

j=2 mj−l−1(1 − q−2 ∑l j=2 mj−l+2 u)

(1 − q−2 ∑l

j=1 mj−l u)(1 − q−2 ∑l j=1 mj−l+2 u)

Ψ+

i, m, 1(u) = qmi−1−mi (1 − q−2 ∑l

j=i−1 mj−l+i−1 u)(1 − q−2 ∑l j=i+1 mj−l+i+1 u)

(1 − q−2 ∑l

j=i mj−l+i−1 u)(1 − q−2 ∑l j=i mj−l+i+1 u)

Therefore Ψ+

1, 0, 1(u) = q−l−1(1 − q−l u)−1,

Ψ+

i, 0, 1(u) = 1,

i = 2, . . . , l

Uq(b+) highest ℓ-weights: a = 2, . . . , l

The ℓ-weights (i = 1, . . . , a − 2, a + 1, . . . , l) Ψ+

a−1, m, a(u) = q∑l−a+1

j=1

mj−∑l−1

j=l−a+2 mj−2ml+l−a+1

1 − q−2 ∑l−a+1

j=1

mj−l+a u

• Ψ+

a, m, a(u) = q−2m1−∑l−a+1

j=2

mj+∑l

j=l−a+2 mj−l+a−2

1 − q−2 ∑l−a+1

j=2

mj−l+a+1 u

• 1 − q−2 ∑l−a+1

j=1

mj−l+a−1 u

1 − q−2 ∑l−a+1

j=1

mj−l+a+1 u

• Ψ+

i, m, a(u) = qmi−a−mi−a+1 (1 − q−2 ∑l−a+1

j=i−a mj−l+i−1 u)(1 − q−2 ∑l−a+1 j=i−a+2 mj−l+i+1 u)

(1 − q−2 ∑l−a+1

j=i−a+1 mj−l+i−1 u)(1 − q−2 ∑l−a+1 j=i−a+1 mj−l+i+1 u)

Therefore Ψ+

i, 0, a(u) =

       1, i = 1, . . . , a − 2 ql−a+1 (1 − q−l+a u), i = a − 1 q−l+a−2 (1 − q−l+a−1 u)−1, i = a 1, i = a + 1, . . . , l

Uq(b+) highest ℓ-weights: a = l + 1

The ℓ-weights Ψ+

i, m, l+1(u) = qmi+1−mi,

i = 1, . . . , l − 1 Ψ+

l, m, l+1(u) = q−2ml−∑l−1

j=1 mj (1 − q u)

Therefore Ψ+

i, 0, l+1(u) = 1,

i = 1, . . . , l − 1, Ψ+

l, 0, l+1(u) = (1 − q u)

Highest ℓ-weights altogether a = 1 Ψ+

i, 0, 1(u) =

• q−l−1(1 − q−l u)−1,

i = 1 1, i = 2, . . . , l a = 2, . . . , l Ψ+

i, 0, a(u) =

       1, i = 1, . . . , a − 2 ql−a+1 (1 − q−l+a u), i = a − 1 q−l+a−2 (1 − q−l+a−1 u)−1, i = a 1, i = a + 1, . . . , l a = l + 1 Ψ+

i, 0, l+1(u) =

• 1,

i = 1, . . . , l − 1 (1 − q u), i = l

Uq(b+) highest ℓ-weights altogether

q-oscillator vs. prefundamental reps θ1 ∼ = Lξ L−

1,q−l

θa ∼ = [Lξ L+

a−1,q−l+a ⊗ Lξ L− a,q−l+a−1],

a = 2, . . . , l θl+1 ∼ = L+

l,q

ξ(ha−1) = l − a + 1, ξ(ha) = −l + a − 2, ξ(hi) = 0, i = a − 1, a ℓ-weights for θa, a = 1, . . . , l + 1 Ψ+

i, m, a(u) = Ψ+ l−i+1, m, l−a+2(−(−1)l u),

i = 1, . . . , l The same holds also for (W1)ζ1 ⊗ · · · ⊗ (Wl+1)ζl+1 and ( Vλ)ζ[ξ] General linear case sll+1 : 2l + 2 Oscq reps vs. 2l prefundamental reps sl2 is special : 2 Oscq vs. 2 prefundamental reps

( Vλ)ζ[ξ] versus l+1

a=1(Wa)ζa

We first note for (W1)ζ1 ⊗ · · · ⊗ (Wl+1)ζl+1 Ψ+

i (u) = q−2 1 − q−l+i+1 ζs i+1 u

1 − q−l+i−1 ζs

i u ,

i = 1, . . . , l Further, restricting ( Vλ)ζ to Uq(b+) Ψ+

i (u) = qλi−λi+1 1 − q2λi+1−i+1 ζs u

1 − q2λi−i+1 ζs u , i = 1, . . . , l In particular Ψ+

1,m(u) = qλ1−λ2−2m12−∑l+1

i=3(m1i−m2i)

×

• 1 − q2λ1−2 ∑l+1

i=3 m1i+2 u

• 1 − q2λ2−2 ∑l+1

i=3 m2i u

• 1 − q2λ1−2 ∑l+1

i=2 m1i u

• 1 − q2λ1−2 ∑l+1

i=2 m1i+2 u

• The comparison gives the isomorphism at

ζi = q2(λ+ρ)i/s ζ, ρi = l 2 − i + 1, i = 1, . . . , l + 1 ξ(hi) = −λi + λi+1 − 2, i = 1, . . . , l

Appendix

Quantum group Uq(gll+1) (q = e¯

h)

Generators and defining relations (κq = q − q−1, A(k) = Ak/[k]q!) Ei, Fi, i = 1, . . . , l, qX, X ∈ kl+1 =

l+1

• i=1

CKi q0 = 1, qX1qX2 = qX1+X2 qXEiq−X = qαi(X)Ei, qXFiq−X = q−αi(X)Fi [Ei, Fj] = δijκ−1

q

• qKi−Ki+1 − qKi+1−Ki
• 1−aij

∑

k=0

(−1)k E(1−aij−k)

i

Ej E(k)

i

= 0,

1−aij

∑

k=0

(−1)k F(1−aij−k)

i

Fj F(k)

i

= 0 All (l + 1)l/2 positive roots (△+) in normal order (α1), (α1 + α2, α2), (α1 + α2 + α3, α2 + α3, α3), . . . . . . (α1 + α2 + . . . + αi, α2 + . . . + αi, . . . , αi), . . . . . . (α1 + α2 + . . . + αl, α2 + . . . + αl, . . . , αl)

Quantum loop algebra Uq(L(sll+1)) (q = e¯

h)

Uq(L(sll+1)) defining relations (q = e¯

h)

{qx, ei, fi}, x ∈ Chl+1, hi ∈ hl+1, αi ∈ h∗

l+1,

i = 0, 1, . . . , l q0 = 1, qx1qx2 = qx1+x2, qν ∑i hi = 1 qxeiq−x = qαi(x)ei, qxfiq−x = q−αi(x)fi [ei, fj] = δijκ−1

q (qhi − q−hi) 1− aij

∑

k=0

(−1)ke(1−

aij−k) i

ej e(k)

i

= 0,

1− aij

∑

k=0

(−1)kf (1−

aij−k) i

fj f (k)

i

= 0 Hopf algebra structure ∆(qx) = qx ⊗ qx, ∆(ei) = ei ⊗ 1 + qhi ⊗ ei, ∆(fi) = fi ⊗ q−hi + 1 ⊗ fi S(qx) = q−x, S(ei) = −q−hiei, S(fi) = −fi qhi ε(qx) = 1, ε(ei) = 0, ε(fi) = 0

Cartan–Weyl generators of Uq(L(sll+1))

System of positive roots of sll+1 (imaginary root δ = α0 + θ)

• △+ = {γ + kδ | γ ∈ △+, k ∈ Z+}

∪ {mδ | m ∈ N} ∪ {(δ − γ) + nδ | γ ∈ △+, n ∈ Z+} Normal order γ + kδ ≺ mδ ≺ (δ − γ) + nδ, γ ∈ △+, k, m, n ∈ Z+ Higher root vectors (eαi = ei, fαi = fi) eγ+nδ = [2]−1

q [eγ+(n−1)δ, e′ δ, γ]q,

fγ+nδ = [2]−1

q [f ′ δ, γ, fγ+(n−1)δ]q

e(δ−γ)+nδ = [2]−1

q [e′ δ, γ, e(δ−γ)+(n−1)δ]q

f(δ−γ)+nδ = [2]−1

q [f(δ−γ)+(n−1)δ, f ′ δ, γ]q

e′

nδ, γ = [eγ+(n−1)δ, eδ−γ]q,

f ′

nδ, γ = [fδ−γ, fγ+(n−1)δ]q

−κq eδ,γ(u) = log(1 − κq e′

δ, γ(u)),

κq fδ,γ(u−1) = log(1 + κq f ′

δ, γ(u−1))

Second Drinfeld’s realization of Uq(L(sll+1))

New set of generators ξ±

i, n,

i = 1, . . . , l, n ∈ Z qx, x ∈ hl+1, χi, n, i = 1, . . . , l, n ∈ Z \ {0} Defining relations q0 = 1, qx1qx2 = qx1+x2 [χi, n, χj, m] = 0, qxχj, n = χj, n qx qxξ±

i, nq−x = q±αi, xξ± i, n,

[χi, n, ξ±

j,m] = ± 1

n[n aij]q ξ±

j, n+m

ξ±

i, n+1ξ± j, m − q±aij ξ± j, m ξ± i, n+1 = q±aij ξ± i, n ξ± j, m+1 − ξ± j, m+1ξ± i, n

[ξ+

i, n, ξ− j, m] = δij κ−1 q

(φ+

i, n+m − φ− i, n+m)

with

∞

∑

n=0

φ±

i, ±nu±n = q±hi exp

• ±κq

∞

∑

n=1

χi, ±nu±n

• φ+

i, n = 0,

n < 0, φ−

i, n = 0,

n > 0

Drinfeld–Jimbo’s versus 2nd Drinfeld’s

Generators ξ±

i, n and χi, n

ξ+

i, n =

• (−1)non

i eαi+nδ

n ≥ 0 (−1)n+1on

i q−hi i

f(δ−αi)−(n+1)δ n < 0 ξ−

i, n =

• (−1)non+1

i

e(δ−αi)+(n−1)δ qhi

i

n > 0 (−1)non

i fαi−nδ

n ≤ 0 χi, n =

• (−1)n+1on

i enδ, αi

n > 0 (−1)n+1on

i f−nδ, αi

n < 0 where oi is either +1 or −1, so that oi = −oj whenever aij < 0 Generators φ±

i, n

φ+

i, n =

• (−1)n+1on

i κq qhi i e′ nδ, αi

n > 0 qhi

i

n = 0 φ−

i, n =

• q−hi

i

n = 0 (−1)non

i κq q−hi i

f ′

−nδ, αi

n < 0 φ+

i (u) = qhi i

• 1 − κq e′

δ, αi(−oiu)

• ,

φ−

i (u−1) = q−hi i

• 1 + κq f ′

δ, αi(−oiu−1)

The weight modules and the category O

A Uq(L(g))-module V is called a weight module if V =

• λ∈

h∗

Vλ, Vλ = {v ∈ V | qxv = qλ, xv for any x ∈ h} A Uq(L(g))-module V is said to be in the category O if

(i) V is a weight module all of whose weight spaces are finite dimensional; (ii) there exists a finite number of elements µ1, . . . , µs ∈ h∗ such that every weight of V belongs to the set

s

• i=1

{µ ∈ h∗ | µ ≤ µi}, where ≤ is the usual partial order in h∗

A Uq(L(g))-module V in the category O is called a highest weight module with highest weight λ if there exists a weight vector v ∈ V of weight λ such that eiv = 0, i ∈ I, and V = Uq(L(g))v Such a vector v is unique up to a scalar factor. It is called the highest weight vector of V.

The Weyl group and quantum B.G.G. resolution

The Weyl group W for gll+1 ri : g∗ → g∗, ri(λ) = λ − λ(Ki − Ki+1) αi, i = 1, . . . , l B.G.G. resolution: Exact sequence of Uq(gll+1)-modules Uk =

• w∈W

ℓ(w)=k

• Vw·λ,

w · λ = w(λ + ρ) − ρ 0 − → Ul+1

ϕl+1

− → Ul

ϕl

− → . . .

ϕ1

− → U0

ϕ0

− → U−1 − → 0, U−1 = Vλ Traces and universal transfer operators trλ = ∑

w∈W

(−1)ℓ(w) tr w·λ Tλ

i (ζ) = ∑ p∈Sl+1

sgn(p) T p(λ+ρ)−ρ

i

(ζ), T′λ

i (ζ) = ∑ p∈Sl+1

sgn(p) T′ p(λ+ρ)−ρ

i

(ζ) Determinant formulae

{i, j=1,...,l+1}

C Tλ−ρ(ζ) = det

• Qi(q−2λj/sζ)
• ,

C Tλ−ρ(ζ) = det

• Qi(q2λj/sζ)

Universal TQ-relations

The universal TQ-relations follow from the identity (∀j=1,...,n≡l+1) det         Q1(q−2λ1/sζ) · · · Q1(q−2λn/sζ) Q1(q−2λn+1/sζ) Q2(q−2λ1/sζ) · · · Q2(q−2λn/sζ) Q2(q−2λn+1/sζ) . . . ... . . . . . . Qn(q−2λ1/sζ) · · · Qn(q−2λn/sζ) Qn(q−2λn+1/sζ) Qj(q−2λ1/sζ) · · · Qj(q−2λn/sζ) Qj(q−2λn+1/sζ)         = 0 Particular cases l = 1 T(λ1−1/2, λ2+1/2)(ζ)Qj(q−2λ3/sζ) − T(λ1−1/2, λ3+1/2)(ζ)Qj(q−2λ2/sζ) +T(λ2−1/2, λ3+1/2)(ζ)Qj(q−2λ1/sζ) = 0 l = 2 T(λ1−1, λ2, λ3+1)(ζ)Qj(q−2λ4/sζ) − T(λ1−1, λ2, λ4+1)(ζ)Qj(q−2λ3/sζ) +T(λ1−1, λ3, λ4+1)(ζ)Qj(q−2λ2/sζ) − T(λ2−1, λ3, λ4+1)(ζ)Qj(q−2λ1/sζ) = 0

Interesting specializations

l = 1 : { λ1=1, λ2=−1, λ3=0 } T(1,−1)(ζ)Qj(ζ) = Qj(q2/sζ) + Qj(q−2/sζ) l = 2 : { λ1=2, λ2=1, λ3=0, λ4=−1 } T(1, 1, 0)(ζ)Qj(ζ) − T(1, 0, 0)(ζ)Qj(q−2/sζ) = Qj(q2/sζ) − Qj(q−4/sζ) T(1, 1, 0)(ζ)Qj(ζ) − T(1, 0, 0)(ζ)Qj(q2/sζ) = Qj(q−2/sζ) − Qj(q4/sζ) Universal TQQ-relations from Jacobi identity T(1,0,0)(ζ)Qi(q−2/sζ)Qj(q−1/sζ) = Qi(q−4/sζ)Qj(q−1/sζ) + Qi(ζ)Qj(q−3/sζ) + Qi(q−2/sζ)Qj(q1/sζ) T(1,1,0)(ζ)Qi(ζ)Qj(q−1/sζ) = Qi(q2/sζ)Qj(q−1/sζ) + Qi(q−2/sζ)Qj(q1/sζ) + Qi(ζ)Qj(q−3/sζ)

Universal TT-relations

The universal TT-relations follow from the identity det         Q1(q−2λ1/sζ) · · · Q1(q−2λn/sζ) Q1(q−2λn+1/sζ) Q2(q−2λ1/sζ) · · · Q2(q−2λn/sζ) Q2(q−2λn+1/sζ) . . . ... . . . . . . Qn(q−2λ1/sζ) · · · Qn(q−2λn/sζ) Qn(q−2λn+1/sζ) T(λ1,λn+2,...,λ2n)(ζ) · · · T(λn,λn+2,...,λ2n)(ζ) T(λn+1,λn+2,...,λ2n)(ζ)         = 0 Particular cases l = 1 T(λ1−1/2, λ2+1/2)(ζ)T(λ3, λ4)(ζ) − T(λ1−1/2, λ3+1/2)(ζ)T(λ2, λ4)(ζ) +T(λ2−1/2, λ3+1/2)(ζ)T(λ1, λ4)(ζ) = 0 l = 2 T(λ1−1, λ2, λ3+1)(ζ)T(λ4, λ5, λ6)(ζ) − T(λ1−1, λ2, λ4+1)(ζ)T(λ3, λ5, λ6)(ζ) +T(λ1−1, λ3, λ4+1)(ζ)T(λ2, λ5, λ6)(ζ) − T(λ2−1, λ3, λ4+1)(ζ)T(λ1, λ5, λ6)(ζ) = 0

Interesting specializations

l = 1 { λ1=ℓ, λ2=0, λ3=ℓ−1, λ4=−1 } T(ℓ,0)(ζ)T(ℓ,0)(q2/sζ) = 1 + T(ℓ−1,0)(ζ)T(ℓ+1,0)(q2/sζ) { λ1=1, λ2=ℓ, λ3=−1, λ4=0 } T(1,−1)(ζ)T(ℓ,0)(ζ) = T(ℓ+1,0)(q2/sζ) + T(ℓ−1,0)(q−2/sζ) l = 2 { λ1=ℓ+2, λ2=ℓ+1, λ3=1, λ4=0, λ5=0, λ6=−1 } T(ℓ−1, 0, 0)(ζ)T(ℓ+1, 0, 0)(q2/sζ) = T(ℓ, 0, 0)(ζ)T(ℓ, 0, 0)(q2/sζ) − T(ℓ, ℓ, 0)(ζ) { λ1=ℓ+2, λ2=ℓ, λ3=0, λ4=ℓ+1, λ5=ℓ+1, λ6=−1 } T(ℓ−1, ℓ−1, 0)(q−2/sζ)T(ℓ+1, ℓ+1, 0)(ζ) = T(ℓ, ℓ, 0)(q−2/sζ)T(ℓ, ℓ, 0)(ζ) − T(ℓ, 0, 0)(ζ)

Quantum Jacobi-Trudi identity

Let ℓ1, ℓ2 ∈ Z>0, ℓ1 ≥ ℓ2

{ λ1=ℓ1+2, λ2=ℓ2+1, λ3=1, λ4=0, λ5=0, λ6=−1 }

T(ℓ1, ℓ2, 0)(ζ) = T(ℓ1, 0, 0)(ζ)T(ℓ2, 0, 0)(q2/sζ) − T(ℓ1+1, 0, 0)(q2/sζ)T(ℓ2−1, 0, 0)(ζ) Tℓ1,ℓ2,0(ζ) can be expressed via Tℓ,0,0(ζ) = ⇒ via T1,0,0(ζ), T1,1,0(ζ) T(ℓ1, ℓ2, 0)(ζ) = det

• Eℓt

i−i+j(q−2(j−1)/sζ)

• 1≤i, j≤ℓ1

E0(ζ) = E3(ζ) = 1, E1(ζ) = T(1, 0, 0)(ζ), E2(ζ) = T(1, 1, 0)(ζ), Ek(ζ) = 0 ∀ k < 0, k > 3 ℓt

i = 2,

1 ≤ i ≤ ℓ2, ℓt

i = 1,

ℓ2 < i ≤ ℓ1

Acting by E1,l+1 on the basis

Subsidiary relations Ei, l+1 F

mij ij

= F

mij ij Ei, l+1 − q−mij+1 [mij]q F mij−1 ij

Ej, l+1 qHij, 1 ≤ i < j ≤ l Ei, l+1 Fmi, l+1

i, l+1 = Fmi, l+1 i, l+1 Ei, l+1 + [mi, l+1]q Fmi, l+1−1 i, l+1

[Hi, l+1 − mi, l+1 + 1]q, 1 ≤ i ≤ l Ej, l+1 Fmik

ik

= Fmik

ik Ej, l+1 − κq [mik]q Fij Fmik−1 ik

Ek, l+1 qHjk, 1 ≤ i < j < k ≤ l Ej, l+1 Fmi, l+1

i, l+1 = Fmi, l+1 i, l+1 Ej, l+1 + [mi, l+1]q Fij Fmi, l+1−1 i, l+1

qHj, l+1, 1 ≤ i < j ≤ l Ei, l+1 F

mj, l+1 j, l+1 = F mj, l+1 j, l+1 Ei, l+1 + qmj, l+1 [mj, l+1]q F mj, l+1−1 j, l+1

Eij q−Hj, l+1, 1 ≤ i < j ≤ l where qνHij = qν ∑

j−1 k=i Hk. Here, in the 1st relation, Ej, l+1 Fij = Fij Ej, l+1 and

qHij Fij = q−2 Fij qHij, Ej, l+1 qHij = q−2 qHij Ej, l+1 More subsidiary relations F

mjk jk Fij = q−mjk Fij F mjk jk + [mjk]q Fik F mjk−1 jk

, 1 ≤ i < j < k ≤ l + 1 F

mjn jn Fik = Fik F mjn jn + κq qmjn−1 [mjn]q Fin Fjk F mjn−1 jn

, 1 ≤ i < j < k < n ≤ l + 1

Continuing with the action of E1,l+1 on the basis

For 0 ≤ k ≤ l − 1 denote Λl, k = {(i0 = 1, i1, . . . , ik, ik+1 = l+ 1) ∈ N×(k+2) | i0 < i1 < . . . < ik < ik+1} The first and last elements of the tuples entering Λl, k are fixed. However, it is convenient to consider the tuples of such form. We denote an element (i0, i1, . . . , ik, ik+1) of Λl, k by i, and Ψl, k = {(i, j) ∈ Λl, k × Λl, k | ja−1 < ia ≤ ja, a = 1, . . . k} Then we write E1, l+1vm =

l−1

∑

k=0

∑

(i, j)∈Ψ

l, k

ik=jk

am|k, i, j vm−ǫj0j1+ǫi1j1−...−ǫjk−1jk +ǫikjk −ǫjkjk+1 +

l−1

∑

k=1

∑

(i, j)∈Ψ

l, k

ik=jk

bm|k, i, j vm−ǫj0j1+ǫi1j1−...−ǫjk−1jk +ǫikjk −ǫjkjk+1 It is assumed here that ǫii = 0.

Continuing with the action of E1,l+1 on the basis

The explicit form of the coefficients am|k, i, j and bm|k, i, j is am|k, i, j = (−1)kκγi, j

q

qλ1−λjk −∑l+1

j=2 m1j+mjk, l+1−∑k a=1 δm|a, i, j+k−γi, j

× [λjk − λl+1 −

l

∑

j=jk+1

mikj −

l

∑

i=ik+1

mi, l+1 + 1]q

k+1

∏

a=1

[mia−1ja]q bm|k, i, j = (−1)kκγi, j−1

q

qλ1−λl+1−∑l+1

j=2 m1j−∑k a=1 δm|a, i, j+k−γi, j

k+1

∏

a=1

[mia−1ja]q where γi, j = ♯({a = 1, . . . , k | ia = ja}) and δm|a, i, j =

ja−1

∑

j=ja−1+1

mia−1j +

ia

∑

i=ia−1+1

mija It is also assumed that mii = 0