quantum a ne gln via hecke algebrasariki/shanghai2015talkfiles/...the hecke algebra h(r) has a...

Quantum affine gln via Hecke algebras

Qiang Fu

Tongji University

Qiang Fu (Tongji University) Quantum affine gln via Hecke algebras 1 / 53

Outline

BLM realization of quantum gln

BLM realization of quantum affine gln

An integral form of quantum affine gln

Outline

Quantum gln

The quantum enveloping algebra of gln is the Q(v) (v an

indeterminant) algebra U(n) presented by generators

Ei, Fi (1 6 i 6 n− 1), Kj , K−1j (1 6 j 6 n) and relations

(a) KiKj = KjKi, KiK−1i = 1;

(b) KiEj = vδi,j−δi,j+1EjKi, KiFj = vδi,j+1−δi,jFjKi;

(c) EiFj − FjEi = δi,jKi−K−1

iv−v−1 , where Ki = KiK

−1i+1;

(d) (quantum Serre relations)

EiEj = EjEi, FiFj = FjFi when |i− j| > 1;

E2i Ej − (v + v−1)EiEjEi + EjE

2i = 0 when |i− j| = 1;

F 2i Fj − (v + v−1)FiFjFi + FjF

2i = 0 when |i− j| = 1.

Hecke algebras

Let Sr be the symmetric group with generators si (1 6 i 6 r − 1),

where si = (i, i+ 1).

The Hecke algebra H(r) of type A over Q(v) is the algebra

generated by Tsi (1 6 i 6 r − 1) with the following relations:

T 2si = (v2 − 1)Tsi + v2, TsiTsj = TsjTsi (i− j 6= ±1),

TsiTsjTsi = TsjTsiTsj (i− j = ±1),

Hecke algebras

Let Sr be the symmetric group with generators si (1 6 i 6 r − 1),

where si = (i, i+ 1).

The Hecke algebra H(r) of type A over Q(v) is the algebra

generated by Tsi (1 6 i 6 r − 1) with the following relations:

T 2si = (v2 − 1)Tsi + v2, TsiTsj = TsjTsi (i− j 6= ±1),

TsiTsjTsi = TsjTsiTsj (i− j = ±1),

q-Schur algebras

Let Ωn be the Q(v)-space with basis ωi16i6n.

The set ωi := ωi1 ⊗ ωi2 ⊗ · · · ⊗ ωir | i ∈ I(n, r) forms a Q(v)-basis

for Ω⊗rn , where I(n, r) = (i1, . . . , ir) ∈ Zr | 1 6 ik 6 n, ∀k.

Ω⊗rn is a right H(r)-module:

ωi · Tsk =

v2ωi, if ik = ik+1;

vωisk , if ik < ik+1; for all i ∈ I(n, r),

vωisk + (v2 − 1)ωi, if ik+1 < ik,

The algebra S(n, r) = EndH(r)(Ω⊗rn ) is called q-Schur algebra.

q-Schur algebras

ωi · Tsk =

q-Schur algebras

ωi · Tsk =

q-Schur algebras

ωi · Tsk =

Quantum Schur-Weyl duality

(Jimbo, 1986) quantum Schur–Weyl duality: a theorem that relates

representation theory of quantum gln and Hecke algebras of type A.

quantum Schur–Weyl duality: two epimorphisms

U(n) S(n, r) = EndH(r)(Ω⊗rn );

H(r) EndU(n)(Ω⊗rn ).

q-Schur algebras are used to bridge representations of quantum gln

and Hecke algebras of type A. The polynomial representation theory

of quantum gln are controlled by the representation theory of q-Schur

algebras.

The Hecke algebra H(r) has a Q(v)-basis Tww∈Sr . The algebraic

structure of H(r) is determined by the following multiplication

formulas, for all 1 6 i 6 r − 1 and w ∈ Sr,

TsiTw =

(v2 − 1)Tw + v2Tsiw, if `(siw) < `(w);

Tsiw, if `(siw) = `(w) + 1,

Using q-Schur algebras, the quantum enveloping algebra U(n) of gln

was reconstructed by Beilinson–Lusztig–MacPherson (Duke Math. J.,

1990). They constructed a Q(v)-basis A(j) | A ∈ Θ±(n), j ∈ Znfor quantum gln and derived explicit multiplication formulas for

EiA(j), FiA(j) and KjA(j).

The Hecke algebra H(r) has a Q(v)-basis Tww∈Sr . The algebraic

structure of H(r) is determined by the following multiplication

formulas, for all 1 6 i 6 r − 1 and w ∈ Sr,

TsiTw =

(v2 − 1)Tw + v2Tsiw, if `(siw) < `(w);

Tsiw, if `(siw) = `(w) + 1,

Using q-Schur algebras, the quantum enveloping algebra U(n) of gln

was reconstructed by Beilinson–Lusztig–MacPherson (Duke Math. J.,

1990). They constructed a Q(v)-basis A(j) | A ∈ Θ±(n), j ∈ Znfor quantum gln and derived explicit multiplication formulas for

EiA(j), FiA(j) and KjA(j).

Using the stabilization property of multiplication for q-Schur algebras,

Beilinson-Lusztig-Macpherson (BLM) construct an algebra K(n)

(without unity) with basis [A] | A ∈ Θ(n), where Θ(n) is the set of

all n× n matrices A = (ai,j) over Z such that ai,j > 0 for i 6= j.

K(n) : a certain completion of K(n). It is an associative algebra with

a unit element:∑

λ∈Zn [diag(λ)].

For A ∈ Θ±(n) := A ∈ Θ(n) | ai,i = 0, ∀i, j ∈ Zn, define

A(j) =∑

λ∈Zn vλ1j1+···+λnjn [A+ diag(λ)](infinite sum) ∈ K(n).

Theorem ([Beilinson–Lusztig–Macpherson, Duke Math.J., 1990])

Let V(n) = spanA(j) | A ∈ Θ±(n), j ∈ Zn ⊆ K(n). Then V(n) is a

subalgebra of K(n) and there is an algebra isomorphism η : U(n)→ V(n)

such that Ei 7→ Ei,i+1(0), Fi 7→ Ei+1,i(0), and Kj11 · · ·K

jnn 7→ 0(j).

a unit element:∑

λ∈Zn [diag(λ)].

A(j) =∑

jnn 7→ 0(j).

a unit element:∑

λ∈Zn [diag(λ)].

A(j) =∑

jnn 7→ 0(j).

a unit element:∑

λ∈Zn [diag(λ)].

A(j) =∑

jnn 7→ 0(j).

Canonical bases of K(n)

Let K(n)Z = spanZ[A] | A ∈ Θ(n), where Z = Z[v, v−1]. There exists

a unique Z-basis θA | A ∈ Θ(n) for K(n)Z such that θA = θA and

θA − [A] ∈∑

B∈Θ(n),B@A v−1Z[v−1][B]. (Canonical basis)

Theorem (Du–Fu, J. Algebra, 2009)

(1) K(n) ∼= U(gln).

(2) There is a surjective algebra homomorphism ζr : K(n)→ S(n, r)

such that ζr([A]) =

[A], if A ∈ Θ(n, r);

0, otherwise.

θA − [A] ∈∑

(1) K(n) ∼= U(gln).

[A], if A ∈ Θ(n, r);

0, otherwise.

θA − [A] ∈∑

(1) K(n) ∼= U(gln).

[A], if A ∈ Θ(n, r);

0, otherwise.

θA − [A] ∈∑

(1) K(n) ∼= U(gln).

[A], if A ∈ Θ(n, r);

0, otherwise.

Theorem (Fu, J. Algebra, 2014)

(1) Let θA,r | A ∈ Θ(n, r) be the canonical basis for q-Schur algebras.

Then we have ζr(θA) =

θA,r, if A ∈ Θ(n, r);

0, otherwise.

(2) Let θ+A | A ∈ Θ+(n) be the canonical basis for the positive part of

U(n). Then we have θ+A =

∑µ∈Zn θA+diag(µ).

0, otherwise.

Applications

(partial integral Schur–Weyl duality) Using BLM’s work, Du proved

that ζr(UZ(n)) = SZ(n, r), where UZ(n) is the integral form of

U(n) and SZ(n, r) is the q-Schur algebra over Z = Z[v, v−1] ([Jie

Du, Alg. Colloq., 1995]).

It is well known that the positive part of U(n) has a canonical basis.

The whole quantum algebra U(n) doesn’t have a canonical basis.

However the algebra K(n) has a canonical basis. Furthermore, the

category of U(n)-modules of type 1 is equivalent to the category of

unital K(n)-modules.

Applications

(partial integral Schur–Weyl duality) Using BLM’s work, Du proved

that ζr(UZ(n)) = SZ(n, r), where UZ(n) is the integral form of

U(n) and SZ(n, r) is the q-Schur algebra over Z = Z[v, v−1] ([Jie

Du, Alg. Colloq., 1995]).

It is well known that the positive part of U(n) has a canonical basis.

The whole quantum algebra U(n) doesn’t have a canonical basis.

However the algebra K(n) has a canonical basis. Furthermore, the

category of U(n)-modules of type 1 is equivalent to the category of

unital K(n)-modules.

Applications

The algebra K(n) was later generalized by Lusztig to other types,

which is called modified quantum groups (see Lusztig’s book

[Progress in Math., 1993]).

BLM realization of Lusztig-Z (Z = Z[v, v−1]) form of quantum gln

([Fu, Commun. Contemp. Math., 2015]).

BLM realization of Frobenius–Lusztig kernel of quantum gln ([Fu,

Math. Res. Lett., to appear]).

Applications

Problem

How to use affine quantum Schur algebra to realize the quantum loop

algebra U(gln)?

U(gln) ∼= the double Ringel–Hall algebra of affine type A.

(Ginzburg–Vasserot Internat. Math. Res. Notices 1993; Lusztig Asian

J. Math. 1999):

geometric definition of the affine quantum Schur algebra SM(n, r).

There is an algebra homomorphism ζr from the quantum loop algebra

of sln to SM(n, r).

(Lusztig) The map ζr is not surjective in the case where n 6 r.

Problem

algebra U(gln)?

J. Math. 1999):

of sln to SM(n, r).

Problem

algebra U(gln)?

J. Math. 1999):

of sln to SM(n, r).

Problem

algebra U(gln)?

J. Math. 1999):

of sln to SM(n, r).

Problem

algebra U(gln)?

J. Math. 1999):

of sln to SM(n, r).

Problem

algebra U(gln)?

J. Math. 1999):

of sln to SM(n, r).

Problem

(Du–Fu, Math. Z., 2010) A realization conjecture of the quantum

loop algebra U(gln) was formulated.

(Deng–Du–Fu, LMS Lecture Note Series 2012, Chapter 6) BLM

realization of the universal enveloping algebra of gln. (The realization

conjecture is true in the case of v = 1.)

(Du–Fu, Adv. Math., 2015) BLM realization of the quantum loop

algebra U(gln).

Problem

algebra U(gln).

Problem

algebra U(gln).

Ringel–Hall algebras of the cyclic quiver

Let Rep04(n) = Rep0F4(n) be the category of finite dimensional

nilpotent representations of the cyclic quiver 4(n) over F.

The isoclasses of representations in Rep04(n) are indexed by Θ+M (n),

where Θ+M (n) is defined as follows.

Let ΘM(n) be the set of all matrices A = (ai,j)i,j∈Z with ai,j ∈ N such

that (a)ai,j = ai+n,j+n for i, j ∈ Z; (b) for every i ∈ Z, both sets

j ∈ Z | ai,j 6= 0 and j ∈ Z | aj,i 6= 0 are finite.

Let Θ+M (n) = A ∈ ΘM(n) | ai,j = 0, ∀i > j.

For A = (ai,j) ∈ Θ+M (n), let M(A) be the corresponding

representation of 4(n) in Rep04(n).

It is well known that for A,B,C ∈ Θ+M (n), there is a polynomial

ϕCA,B ∈ Z[v2] such that, for any finite field Fq, ϕCA,B|v2=q is equal to

the number of submodules N of MFq(C) satisfying N ∼= MFq(B) and

MFq(C)/N ∼= MFq(A).

Let HM(n) be the (twisted) Ringel–Hall algebra of the cyclic quiver

4(n), which is a Q(v)-space with basis uA = u[M(A)] | A ∈ Θ+M (n).

The multiplication is given by

uAuB = v〈d(A),d(B)〉∑C∈Θ+

M (n) ϕCA,BuC .

It is well known that for A,B,C ∈ Θ+M (n), there is a polynomial

ϕCA,B ∈ Z[v2] such that, for any finite field Fq, ϕCA,B|v2=q is equal to

the number of submodules N of MFq(C) satisfying N ∼= MFq(B) and

Let HM(n) be the (twisted) Ringel–Hall algebra of the cyclic quiver

4(n), which is a Q(v)-space with basis uA = u[M(A)] | A ∈ Θ+M (n).

The multiplication is given by

uAuB = v〈d(A),d(B)〉∑C∈Θ+

M (n) ϕCA,BuC .

Extended Ringel–Hall algebras

The Ringel–Hall algebra is not a bialgebra. However it can be

extended to a Hopf algebra, which is called extended Ringe–Hall

algebra.

The extended Ringe–Hall algebra HM(n)>0 is the Q(v)-space with

basis u+AKα | α ∈ ZI, A ∈ Θ+

M (n). We have

Kαu+A = v〈d(A),α〉u+

AKα, for all α ∈ ZI, A ∈ Θ+M (n),

KαKβ = Kα+β, for all α, β ∈ ZI.

Dually, we may define the Hopf algebra HM(n)60 with basis

Kαu−A | α ∈ ZI, A ∈ Θ+

M (n).

algebra.

M (n). We have

M (n).

algebra.

M (n). We have

M (n).

Double Ringel–Hall algebras associated with cyclic quivers

Proposition (Xiao)

The Q(v)-bilinear form ψ : HM(n)>0 ×HM(n)60 → Q(v) defined by

ψ(u+AKα,Kβu

−B) = vαβ−〈d(A),d(A)+α〉+2d(A)a−1

A δA,B where α, β ∈ ZI and

A,B ∈ Θ+M (n), is a skew-Hopf pairing.

(HM(n)>0, HM(n)60, ψ) gives rise to the Drinfeld double

DM(n) := D(HM(n)>0,HM(n)60).

The reduced Drinfeld double:

DM(n) = DM(n)/〈1⊗Kα −Kα ⊗ 1 | α ∈ ZI〉.

The set u+AK

j11 · · ·K

jnn u−B | A,B ∈ Θ+

M (n), j ∈ Zn forms a

Q(v)-basis for DM(n).

DM(n) is isomorphic to the quantum loop algebra U(gln).

Proposition (Xiao)

ψ(u+AKα,Kβu

−B) = vαβ−〈d(A),d(A)+α〉+2d(A)a−1

DM(n) := D(HM(n)>0,HM(n)60).

DM(n) = DM(n)/〈1⊗Kα −Kα ⊗ 1 | α ∈ ZI〉.

The set u+AK

j11 · · ·K

Proposition (Xiao)

ψ(u+AKα,Kβu

−B) = vαβ−〈d(A),d(A)+α〉+2d(A)a−1

DM(n) := D(HM(n)>0,HM(n)60).

DM(n) = DM(n)/〈1⊗Kα −Kα ⊗ 1 | α ∈ ZI〉.

The set u+AK

j11 · · ·K

Proposition (Xiao)

ψ(u+AKα,Kβu

−B) = vαβ−〈d(A),d(A)+α〉+2d(A)a−1

DM(n) := D(HM(n)>0,HM(n)60).

DM(n) = DM(n)/〈1⊗Kα −Kα ⊗ 1 | α ∈ ZI〉.

The set u+AK

j11 · · ·K

Proposition (Xiao)

ψ(u+AKα,Kβu

−B) = vαβ−〈d(A),d(A)+α〉+2d(A)a−1

DM(n) := D(HM(n)>0,HM(n)60).

DM(n) = DM(n)/〈1⊗Kα −Kα ⊗ 1 | α ∈ ZI〉.

The set u+AK

j11 · · ·K

Presentation of DM(n)

The algebra DM(n) is the Q(v)-algebra generated by

Ei = u+i , Fi = u−i , Ki, K

−1i , z+

s , z−s , i ∈ I, s ∈ Z+ with relations

(i, j ∈ I and s, t ∈ Z+):

(1) KiKj = KjKi, KiK−1i = 1;

(2) KiEj = vδi,j−δi,j+1EjKi, KiFj = v−δi,j+δi,j+1FjKi;

(3) EiFj − FjEi = δi,jKi−K−1

iv−v−1 , where Ki = KiK

−1i+1;

(4)∑

a+b=1−ci,j

(−1)a[

1− ci,ja

]Eai EjE

bi = 0, for i 6= j;

(5)∑

a+b=1−ci,j

(−1)a[

1− ci,ja

]F ai FjF

bi = 0, for i 6= j;

(6) z±s are central elements.

Affine quantum Schur algebras

Let SM,r be the group consisting of all permutations w : Z→ Z such

that w(i+ r) = w(i) + r for i ∈ Z.

The extended affine Hecke algebra HM(r) over Q(v) associated to SM,r

is the algebra with basis Tww∈SM,r , and multiplication defined byT 2si = (v2 − 1)Tsi + v2, for 1 6 i 6 r

TwTw′ = Tww′ , if `(ww′) = `(w) + `(w′).

Affine quantum Schur algebras:

SM(n, r) := EndHM(r)(⊕

λ∈ΛM(n,r)xλHM(r)

where ΛM(n, r) = λ ∈ NnM |∑

16i6n λi = r, xλ =∑

w∈Sλ Tw and

Sλ is the standard Young subgroup of Sr.

TwTw′ = Tww′ , if `(ww′) = `(w) + `(w′).

16i6n λi = r, xλ =∑

w∈Sλ Tw and

TwTw′ = Tww′ , if `(ww′) = `(w) + `(w′).

16i6n λi = r, xλ =∑

w∈Sλ Tw and

Bases of Affine quantum Schur algebras

(Varagnolo–Vasserot) There is a bijective map

M : (λ, d, µ) | d ∈ DMλ,µ, λ, µ ∈ ΛM(n, r) −→ ΘM(n, r).

Here ΘM(n, r) = A ∈ ΘM(n) |∑

16i6n, j∈Z ai,j = r,DMλ = d | d ∈ SM,r, `(wd) = `(w) + `(d) for w ∈ Sλ and

DMλ,µ = DMλ ∩DMµ−1.

For λ, µ ∈ ΛM(n, r) and d ∈ DMλ,µ satisfying A = M(λ, d, µ) ∈ ΘM(n, r),

define eA ∈ SM(n, r) by

eA(xνh) = δµν∑

w∈SλdSµ

where ν ∈ ΛM(n, r) and h ∈ HM(r).

The basis of SM(n, r): eA | A ∈ ΘM(n, r).

w∈SλdSµ

The basis of SM(n, r): eA | A ∈ ΘM(n, r).

w∈SλdSµ

The basis of SM(n, r): eA | A ∈ ΘM(n, r).Qiang Fu (Tongji University) Quantum affine gln via Hecke algebras 21 / 53

The extension of ζr

ζr : U(sln)→ SM(n, r).

Let Ω be the Q(v)-space with basis ωi | i ∈ Z. Ω⊗r admits a right

HM(r)-module structure.

(Varagnolo–Vasserot) SM(n, r) ∼= EndHM(r)(Ω⊗r).

The algebra DM(n) acts on Ω by Ei · ωs = δi+1,sωs−1,

Fi · ωs = δi,sωs+1, K±1i · ωs = v±δi,sωs and z±m · ωs = ωs∓mn.

The tensor space Ω⊗r is a DM(n)-module via the comultiplication ∆

on DM(n).

Since the HM(r)-action commutes with the action of DM(n), we

obtain an algebra homomorphism

ζr : DM(n)→ EndHM(r)(Ω⊗r) = SM(n, r).

on DM(n).

Let [A] = v−dAeA, where dA =∑

16i6ni>k,j<l

ai,jak,l.

Let Θ±M (n) = A ∈ ΘM(n) | ai,i = 0 for all i. For A ∈ Θ±M (n) and

j ∈ ZnM , let

A(j, r) =∑

µ∈ΛM(n,r−σ(A)) vµ1j1+···+µnjn [A+ diag(µ)] ∈ SM(n, r).

Theorem (Deng–Du–Fu, LMS Lecture Note Series 2012, Chapter 3)

The Q(v)-algebra homomorphism ζr : U(gln) = DM(n) SM(n, r) is

surjective. Furthermore, we have

ζr(Kj11 · · ·K

jnn ) = 0(j, r), ζr(u

+A) = A(0, r), and ζr(u

−A) = (tA)(0, r),

where u±A = vd′Au±A and d′A = dim End(M(A))− dimM(A).

16i6ni>k,j<l

ai,jak,l.

j ∈ ZnM , let

A(j, r) =∑

ζr(Kj11 · · ·K

jnn ) = 0(j, r), ζr(u

+A) = A(0, r), and ζr(u

−A) = (tA)(0, r),

16i6ni>k,j<l

ai,jak,l.

j ∈ ZnM , let

A(j, r) =∑

ζr(Kj11 · · ·K

jnn ) = 0(j, r), ζr(u

+A) = A(0, r), and ζr(u

−A) = (tA)(0, r),

The realization problem

Let A(j) := (A(j, r))r>0 ∈∏r>0 SM(n, r),

Let VM(n) = spanA(j) | A ∈ Θ±M (n), j ∈ ZnM ⊆∏r>0 SM(n, r). It

was conjectured in [Du–Fu, Math. Z., 2010] that VM(n) is a

Q(v)-subalgebra of∏r>0 SM(n, r).

ζ :=∏r>0 ζr : U(gln) = DM(n)→

∏r>0 SM(n, r). The map ζ is

injective. This implies that U(gln) = DM(n) ∼= Im(ζ).

If the realization conjecture is true, then Im(ζ) = VM(n) and hence

VM(n) ∼= DM(n) = U(gln).

The realization conjecture is true in the case of v = 1.

Let A(j) := (A(j, r))r>0 ∈∏r>0 SM(n, r),

ζ :=∏r>0 ζr : U(gln) = DM(n)→

Let A(j) := (A(j, r))r>0 ∈∏r>0 SM(n, r),

ζ :=∏r>0 ζr : U(gln) = DM(n)→

Let A(j) := (A(j, r))r>0 ∈∏r>0 SM(n, r),

ζ :=∏r>0 ζr : U(gln) = DM(n)→

Let A(j) := (A(j, r))r>0 ∈∏r>0 SM(n, r),

ζ :=∏r>0 ζr : U(gln) = DM(n)→

Generators of Ringel–Hall algebras

HM(n) = spanuA | A ∈ Θ+M (n) (Ringel–Hall algebras).

The algebra HM(n) is generated by uA with M(A) being semisimple.

The algebra HM(n) is generated by uEMi,i+1and uEMi,i+mn

(1 6 i 6 n, m > 0).

Multiplication formulas in affine Schur algebras

Proposition (Deng–Du–Fu, LMS Lecture Note Series 2012,Chapter 6)

Assume m 6= 0, B ∈ ΘM(n, r) and λ = ro(B). Then in SM(n, r)Q

[EMh,h+mn + diag(λ− eMh )]1[B]1 =∑s∈Z

bh,s>1

(bh,s+mn + 1)[B + EMh,s+mn − EMh,s]1.

The matrix EMh,h+mn corresponds to the imaginary root of affine type

An−1.

The module M(EMh,h+mn) (m > 0) is an indecomposable module of

the cyclic quiver.

bh,s>1

An−1.

the cyclic quiver.

bh,s>1

An−1.

the cyclic quiver.

Proposition (Fu, arXiv:1204.3142.)

Let A ∈ ΘM(n, r) and α, γ ∈ NnM . Assume B ∈ ΘM(n, r) is such that

co(B) = ro(A) and B −∑

16i6nαiE

Mi,i+1 is diagonal and C ∈ ΘM(n, r) is such that

co(C) = ro(A) and C −∑

16i6nγiE

Mi+1 is diagonal. Then in SM(n, r)Z

(1) [B]1[A]1 =∑

T∈ΘM(n)ro(T )=α

∏16i6nj∈Z

(ai,j − ti−1,j + ti,j

)[A+ T − T ]1;

(2) [C]1[A]1 =∑

T∈ΘM(n)ro(T )=γ

∏16i6nj∈Z

(ai,j + ti−1,j − ti,j

ti−1,j

)[A− T + T ]1.

Here ˜ : ΘM(n)→ ΘM(n) A = (ai,j) 7−→ A = (ai−1,j).

The module M(∑

16i6n αiEMi,i+1) is semisimple.

Proposition (Fu, arXiv:1204.3142.)

16i6nαiE

16i6nγiE

Mi+1 is diagonal. Then in SM(n, r)Z

(1) [B]1[A]1 =∑

T∈ΘM(n)ro(T )=α

∏16i6nj∈Z

(ai,j − ti−1,j + ti,j

)[A+ T − T ]1;

(2) [C]1[A]1 =∑

T∈ΘM(n)ro(T )=γ

∏16i6nj∈Z

(ai,j + ti−1,j − ti,j

ti−1,j

)[A− T + T ]1.

Here ˜ : ΘM(n)→ ΘM(n) A = (ai,j) 7−→ A = (ai−1,j).

The module M(∑

16i6n αiEMi,i+1) is semisimple.

Quantized multiplication formula

Proposition (Du–Fu, Adv. Math., 2015)

16i6nαiE

16i6nγiE

Mi+1,i is diagonal. Then in SM(n, r)Z

(1) [B][A] =∑

T∈ΘM(n)ro(T )=α

vβ(T,A)∏

16i6nj∈Z

[[ai,j + ti,j − ti−1,j

]][A+ T − T ],

where β(T,A) =∑

16i6n, j>l(ai,j − ti−1,j)ti,l −∑

16i6n, j>l(ai+1,j − ti,j)ti,l.

(2) [C][A] =∑

T∈ΘM(n)ro(T )=γ

vβ′(T,A)

∏16i6nj∈Z

[[ai,j − ti,j + ti−1,j

ti−1,j

]][A− T + T ],

where β′(T,A) =∑

16i6n, l>j(ai,j − ti,j)ti−1,l −∑

16i6n, l>j(ai,j − ti,j)ti,l.

The triangular relation

Let A ∈ Θ±M (n). Then we have:

A+(0, r)A−(0, r) = A(0, r) +∑

B∈Θ±M (n)

B≺A,j∈ZnM

gB,j,A;rB(j, r) (in SM(n, r)),

where gB,j,A;r ∈ Q(v).

Theorem (Du–Fu, Adv. Math., 2015)

(1) Let A(j) = (A(j, r))r>0 ∈∏r>0 SM(n, r). We have

A+(0)0(j)A−(0) = vj(co(A+)+ro(A−))A(j)+∑

B∈Θ±M (n)

B≺A, j′∈NnM

Q(v)B(j′) (in∏r>0

SM(n, r)),

In particular, gB,j,A;r is independent of r.

(2) Let VM(n) = spanA(j) | A ∈ Θ±M (n), j ∈ ZnM. Then VM(n) is a

Q(v)-subalgebra and U(gln) = DM(n) ∼= VM(n). Furthermore in

VM(n),

(a) 0(j′)A(j) = vj′ro(A)A(j′ + j) and A(j)0(j′) = vj

′co(A)A(j′ + j);

A+(0)0(j)A−(0) = vj(co(A+)+ro(A−))A(j)+∑

B∈Θ±M (n)

B≺A, j′∈NnM

SM(n, r)),

VM(n),

′co(A)A(j′ + j);

A+(0)0(j)A−(0) = vj(co(A+)+ro(A−))A(j)+∑

B∈Θ±M (n)

B≺A, j′∈NnM

SM(n, r)),

VM(n),

′co(A)A(j′ + j);

Theorem (Continue)

(b) (∑

αiEMi,i±1)(0)A(j) =

∑T∈ΘM(n)ro(T )=α

vfT∏

16i6nj∈Z, j 6=i

]](A+ T± − T±)(jT , δT ), where

jT = j +∑

16i6n(∑

j<i(ti,j − ti−1,j))eMi and

fT =∑

16i6nj>l, j 6=i

ai,jti,l −∑

16i6nj>l, j 6=i+1

ai+1,jti,l −∑

16i6nj>l, j 6=i

ti−1,jti,l +∑

16i6nj>l, j 6=i, j 6=i+1

ti,jti,l

16i6nj<i+1

ti,jti+1,i+1 +∑

ji(ti−1,i − ti,i);

where δT = (ti,i)i∈Z ∈ ZnM and T = (ti,j), with ti,j = ti−1,j for all

i, j ∈ Zn.Qiang Fu (Tongji University) Quantum affine gln via Hecke algebras 31 / 53

Theorem (Continue)

(c) (∑

αiEMi±1,i)(0)A(j) =

∑T∈ΘM(n)ro(T )=α

vf′T

∏16i6nj∈Z, j 6=i

[[ai,j − ti,j + ti−1,j

ti−1,j

]](A− T± + T±)(j′T , δT ), where

j′T = j +∑

16i6n(∑

j>i(ti−1,j − ti,j))eMi and

f ′T =∑

16i6nl>j, j 6=i

ai,jti−1,l −∑

16i6nl>j, j 6=i

ai,jti,l −∑

16i6nj>l, l 6=i

ti−1,jti,l +∑

16i6nj>l, l6=i, l 6=i+1

ti,jti,l

16i6ni<j

ti,jti−1,i +∑

ji(ti,i − ti−1,i).

Hall polynomials

ϕCA,B ∈ Z[v2](A,B,C ∈ Θ+M (n)) : Hall polynomials.

For any finite field Fq, ϕCA,B|v2=q is equal to the number of

submodules N of MFq(C) satisfying N ∼= MFq(B) and

The Ringel–Hall algebra HM(n)Z is generated by uSα | α ∈ NnM and whose

(untwisted) multiplication is given by the formulas: for any A ∈ Θ+M (n) and

α ∈ NnM , uSα uA =∑T∈Θ+

M (n), ro(T )=α ϕA+T−T+

Sα,AuA+T−T+ , where

Sα =∑

16i6n αiEMi,i+1 and

ϕA+T−T+

Sα,A= q

∑16i6n, l<j(ai,jti,l−ti,jti+1,l)

∏16i6nj∈Z, j 6=i

Hall polynomials

Sα =∑

ϕA+T−T+

Sα,A= q

∏16i6nj∈Z, j 6=i

Hall polynomials

Sα =∑

ϕA+T−T+

Sα,A= q

∏16i6nj∈Z, j 6=i

Integral Schur–Weyl duality

Integral Schur–Weyl duality of type A: Carter–Lusztig, Math. Z.,

1974; Concini–Procesi, Adv. Math., 1976.

Integral quantum Schur–Weyl duality of type A: Du, Alg. Colloq.,

1995; Du–Parshall–Scott, Comm. Math. Phys., 1998.

Schur–Weyl duality for the symplectic group over arbitrary infinite

field: Concini–Procesi, 1976, Adv. Math.; S. Oehms, J. Algebra,

2001; Dipper–Doty–Hu, Trans. Amer. Math. Soc. 2008.

Schur–Weyl duality for orthogonal groups over an arbitrary infinite

field: Concini–Procesi, Adv. Math., 1976; Doty–Hu, Proc. London

Math. Soc., 2009.

Integral quantum Schur–Weyl duality of type C: J. Hu, Represent.

Theory, 2011.

Math. Soc., 2009.

Theory, 2011.

Math. Soc., 2009.

Theory, 2011.

Math. Soc., 2009.

Theory, 2011.

Math. Soc., 2009.

Theory, 2011.

Problem

Probelm: Integral Schur–Weyl duality of affine type A?

(Deng–Du–Fu, Chapter 3) There is a surjective Q(v)-algebra

homomorphism ζr : DM(n) SM(n, r) (over Q(v)).

Problem: Integral version of the above theorem?

What is a suitable integral form for quantum affine gln?

Let DM(n)Z = spanZu+A

∏16i6nK

[Ki;0λi

]u−B | A,B ∈ Θ+

M (n), j ∈ZnM , λ ∈ NnM, where Z = Z[v, v−1] (v being an indeterminate) and[Ki;0t

]=∏ts=1

Kiv−s+1−K−1

i vs−1

vs−v−s .

It is conjectured in [Deng–Du–Fu] that DM(n)Z is a Z-subalgebra of

quantum affine gln.

Problem

∏16i6nK

[Ki;0λi

]u−B | A,B ∈ Θ+

]=∏ts=1

Kiv−s+1−K−1

i vs−1

vs−v−s .

quantum affine gln.

Problem

∏16i6nK

[Ki;0λi

]u−B | A,B ∈ Θ+

]=∏ts=1

Kiv−s+1−K−1

i vs−1

vs−v−s .

quantum affine gln.

Problem

∏16i6nK

[Ki;0λi

]u−B | A,B ∈ Θ+

]=∏ts=1

Kiv−s+1−K−1

i vs−1

vs−v−s .

quantum affine gln.

Problem

∏16i6nK

[Ki;0λi

]u−B | A,B ∈ Θ+

]=∏ts=1

Kiv−s+1−K−1

i vs−1

vs−v−s .

quantum affine gln.

Commutator relations

In the algebra U(gln) we have:

E(k)i F

(l)i =

∑06t6min(k,l) F

(l−t)i

[Ki;2t−k−l

(k−t)i .

According to this formulas we conclude that the Lusztig Z-form for

U(gln) is a Z-subalgebra of U(gln).

However, the affine case is very complicated since the integral Ringel

Hall algebra D+M (n)Z of affine type A is generated by all u+

λ for

λ ∈ NnM . Here u+λ corresponds to the semisimple module ⊕16i6nλiSi

of the cyclic quiver 4(n).

E(k)i F

(l)i =

∑06t6min(k,l) F

(l−t)i

[Ki;2t−k−l

(k−t)i .

λ for

E(k)i F

(l)i =

∑06t6min(k,l) F

(l−t)i

[Ki;2t−k−l

(k−t)i .

λ for

For λ, µ ∈ NnM , u−µ u+λ − u

+λ u−µ =

∑α 6=0, α∈NnMα6λ, α6µ

∑06γ6α

xα,γK2γ−αu+

λ−αu−µ−α,

xα,γ = v〈α,λ−α〉+〈µ,2γ−α〉+2〈γ,α−γ−λ〉+2σ(α)

α− γ, λ− α, γ

]] [[µ

α− γ, µ− α, γ

]]× aα−γaλ−αaµ−α

aλaµ

m>1,γ(i) 6=0 ∀iγ(1)+···+γ(m)=γ

(−1)mv2∑i<j〈γ

(i),γ(j)〉aγ(1) · · · aγ(m)

γ(1), . . . , γ(m)

with aβ =

n∏i=1

βi∏s=1

(v2βi − v2(s−1)) and Kν := (K1)ν1 · · · (Kn)νn with

Ki = KiK−1i+1 for ν ∈ ZnM .

A integral form of modified quantum affine gln

DM(n) : the modified quantum affine algebra associated with DM(n).

The category of DM(n)-modules of type 1 is equivalent to the

category of unital DM(n)-modules.

DM(n)Z : is a certain Z-submodule of DM(n), which can be regarded

as the modified version of the Z-module DM(n)Z . (Z = Z[v, v−1])

Theorem (Fu, Adv. Math., 2013)

(1) The Z-module DM(n)Z is a Z-subalgebra of DM(n) and the natural

algebra homomorphism ζr : DM(n)Z → SM(n, r)Z is surjective, where

SM(n, r)Z is the affine quantum Schur algebra over Z = Z[v, v−1].

(2) The conjecture of integral form is true in the case of v = 1.

A different approach

Note that the fact that DM(n)Z is a Z-subalgebra of DM(n) doesn’t

imply that DM(n)Z is a Z-subalgebra of DM(n). We have to take a

different route to study the conjecture of integral form.

1. BLM realization of UZ(gln) was given in [Fu, arXiv:1204.3142].

Here, UZ(gln) is a certain Z-submodule of the universal enveloping

algebra of gln.

2. BLM realization of Lusztig Z-form of quantum gln was given in

[Fu, Commun. Contemp. Math., 2015].

3. BLM realization of the Z-module DM(n)Z . In particular, this

implies that the conjecture of integral form is true. ([Du–Fu,

arXiv:1404.5679])

algebra of gln.

arXiv:1404.5679])

algebra of gln.

arXiv:1404.5679])

algebra of gln.

arXiv:1404.5679])

algebra of gln.

arXiv:1404.5679])

BLM realization of UZ(gln)

UZ(gln): a certain Z-submodule of U(gln), where UZ(gln) is the

universal enveloping of gln. In fact, UZ(gln) is a Z-subalgebra of

U(gln) and we have an surjective algebra homomorphism

ηr : UZ(gln)→ SM(n, r)Z (Fu, Adv. Math. 2013).

We introduce the following new elements in affine Schur algebras:

Aj, r =∑

µ∈ΛM(r−σ(A))

)[A+ diag(µ)]1 ∈ SM(n, r)Q.

Let Aj = (Aj, r)r>0 ∈∏r>0 SM(n, r)Q.

Let VM(n)Z = spanZAj | A ∈ Θ±M (n), j ∈ NnM.η : U(gln)→

∏r>0 SM(n, r)Q is an injective algebra homomorphism.

Theorem (Fu, arXiv:1204.3142)

VM(n)Z is a Z-subalgebra of∏r>0 SM(n, r)Q and η(UZ(gln)) = VM(n)Z.

Hence UZ(gln) is a Z-subalgebra of U(gln).

Aj, r =∑

µ∈ΛM(r−σ(A))

Aj, r =∑

µ∈ΛM(r−σ(A))

Aj, r =∑

µ∈ΛM(r−σ(A))

Let VM(n)Z = spanZAj | A ∈ Θ±M (n), j ∈ NnM.

η : U(gln)→∏r>0 SM(n, r)Q is an injective algebra homomorphism.

Aj, r =∑

µ∈ΛM(r−σ(A))

Aj, r =∑

µ∈ΛM(r−σ(A))

Hence UZ(gln) is a Z-subalgebra of U(gln).Qiang Fu (Tongji University) Quantum affine gln via Hecke algebras 40 / 53

BLM realization of Lusztig-Z form of quantum gln

Recall A(j, r) =∑

µ∈Λ(n,r−σ(A)) vµj[A+ diag(µ)] in q-Schur algebras

(BLM).

We introduce the following new elements in q-Schur algebras:

A(j, λ, r) =∑

µ∈Λ(n,r−σ(A)) vµj[µλ

][A+ diag(µ)] ∈ S(n, r).

A(j, r) = A(j,0, r).

Let A(j, λ) = (A(j, λ, r))r>0 ∈∏r>0 S(n, r).

Let V(n)Z = spanZA(j, λ) | A ∈ Θ±(n), j ∈ Zn, λ ∈ Nn.

Theorem (Fu, Commun. Contemp. Math., 2015)

V(n)Z is a Z-subalgebra of∏r>0 S(n, r) and V(n)Z is isomorphic to the

Lusztig-Z form of quantum gln.

Recall A(j, r) =∑

(BLM).

A(j, λ, r) =∑

][A+ diag(µ)] ∈ S(n, r).

A(j, r) = A(j,0, r).

Recall A(j, r) =∑

(BLM).

A(j, λ, r) =∑

][A+ diag(µ)] ∈ S(n, r).

A(j, r) = A(j,0, r).

Recall A(j, r) =∑

(BLM).

A(j, λ, r) =∑

][A+ diag(µ)] ∈ S(n, r).

A(j, r) = A(j,0, r).

Recall A(j, r) =∑

(BLM).

A(j, λ, r) =∑

][A+ diag(µ)] ∈ S(n, r).

A(j, r) = A(j,0, r).

Recall A(j, r) =∑

(BLM).

A(j, λ, r) =∑

][A+ diag(µ)] ∈ S(n, r).

A(j, r) = A(j,0, r).

Recall A(j, r) =∑

(BLM).

A(j, λ, r) =∑

][A+ diag(µ)] ∈ S(n, r).

A(j, r) = A(j,0, r).

BLM realization of a Z-form of quantum affine gln

Let A(j, λ, r) =∑

µ∈ΛM(n,r−σ(A)) vµj[µλ

][A+ diag(µ)] ∈ SM(n, r).

Let A(j, λ) = (A(j, λ, r))r>0 ∈∏r>0 SM(n, r).

Let VM(n)Z = spanZA(j, λ) | A ∈ Θ±M (n), j ∈ ZnM , λ ∈ NnM.

Proposition (Du–Fu, arXiv:1404.5679)

The Z-module VM(n)Z is a subalgebra of∏r>0 SM(n, r) which is generated

by the elements (∑

16i6n αiEMi,i+1)(0), (

∑16i6n αiE

Mi+1,i)(0), 0(eMi ),

0(0, teMi ) for all α ∈ NnM , t ∈ N, 1 6 i 6 n.

16i6n αiEMi,i+1)(0), (

∑16i6n αiE

Mi+1,i)(0), 0(eMi ),

16i6n αiEMi,i+1)(0), (

∑16i6n αiE

Mi+1,i)(0), 0(eMi ),

16i6n αiEMi,i+1)(0), (

∑16i6n αiE

Mi+1,i)(0), 0(eMi ),

ζr : U(gln) = DM(n)→ SM(n, r), ζ : DM(n)→∏r>0 SM(n, r).

ζ is injective and ζ(DM(n)) = VM(n).

Theorem (Du–Fu, arXiv:1404.5679)

(1) We have ζ(DM(n)Z) = VM(n)Z . Thus, DM(n)Z is a subalgebra of

DM(n) and DM(n)Z ∼= VM(n)Z . Moreover, DM(n)Z is a Hopf

subalgebra of DM(n).

(2) The restriction of ζr to DM(n)Z gives a surjective Z-algebra

homomorphism ζr : DM(n)Z SM(n, r)Z . Thus the map ζr is

surjective for any filed k .

The positivity property

The affine quantum Schur algebras play an important role in the

proof of the above theorem. It is also very useful for the investigation

of the positivity property of modified quantum groups.

(Lusztig) The structure constants of the multiplication and

comultiplication for the canonical basis of U(sln)+ are in N[v, v−1].

Using the theory of affine quantum Schur algebras, we can show that

the structure constants for the multiplication and comultiplication

with respect to the canonical basis of modified quantum affine sln

belong to N[v, v−1] for all n if and only if the structure constants for

the multiplication and comultiplication with respect to the canonical

basis of U(sln)+ belong to N[v, v−1] for all n. (Fu–Shoji,

arXiv:1407.4228; Fu, arXiv:1511.05745)

Stabilization Properties of affine quantum Schur algebras

Let Z1 be the subring of Q(v)[v′, v′−1], generated (over Z!) by∏16i6t

v−2(a−i)v′2−1v−2i−1

16i6tv2(a−i)v′−2−1

v2i−1, and vj for all a ∈ Z, t > 1

and j ∈ Z.

ΘM(n) is the set of all matrices A = (ai,j)i,j∈Z with ai,j ∈ N for i 6= j

and ai,i ∈ Z such that (a)ai,j = ai+n,j+n for i, j ∈ Z; (b) for every

i ∈ Z, both sets j ∈ Z | ai,j 6= 0 and j ∈ Z | aj,i 6= 0 are finite.

Let A,B ∈ ΘM(n) and assume co(B) = ro(A). Then there exist unique

X1, · · · , Xm ∈ ΘM(n), unique P1(v, v′), · · · , Pm(v, v′) ∈ Z1 and an integer

p0 > 0 such that, in SM(n, pn+ σ(A))Z ,

[pB][pA] =∑

16i6m Pi(v, v−p)[pXi] for all p > p0.

v−2(a−i)v′2−1v−2i−1

16i6tv2(a−i)v′−2−1

and j ∈ Z.

[pB][pA] =∑

v−2(a−i)v′2−1v−2i−1

16i6tv2(a−i)v′−2−1

and j ∈ Z.

[pB][pA] =∑

BLM realization of modified quantum affine gln

Let KM(n)Z1 be the free Z1-module with basis A | A ∈ ΘM(n).KM(n)Z1 : an associative Z1-algebra (without unit). The structure

constants of KM(n)Z1 are Pi(v, v′).

Let KM(n)Z = KM(n)Z1 ⊗Z1 Z (specializing v′ to 1). The set

·[A]· := A⊗ 1 | A ∈ ΘM(n) forms a Z-basis for KM(n)Z .

Let KM(n) = KM(n)Z ⊗Z Q(v). Then KM(n) ∼= DM(n) and

KM(n)Z ∼= DM(n)Z .

The involution ¯ : SM(n, r)Z → SM(n, r)Z

Let ¯ : HM(r)→ HM(r) be the ring involution defined by v = v−1 and

Tw = T−1w−1 .

We define a map ¯ : SM(n, r)Z → SM(n, r)Z such that v = v−1 and

f(C ′w0,µh) = f(C ′w0,µ)h for f ∈ HomHM(r)(xµHM(r), xλHM(r)) and

h ∈ HM(r), where C ′w0,µ = v−`(w0,µ)xµ.

The map ¯ : SM(n, r)Z → SM(n, r)Z is a ring involution.

Tw = T−1w−1 .

The order relation v

For A ∈ ΘM(n) and i 6= j ∈ Z, let

σi,j(A) =

s6i,t>jas,t, if i < j;∑

s>i,t6jas,t, if i > j.

For A,B ∈ ΘM(n), define

B 4 A if and only if σi,j(B) 6 σi,j(A) for all i 6= j.

Put B ≺ A if B 4 A and, for some pair (i, j) with i 6= j,

σi,j(B) < σi,j(A).

For A,B ∈ ΘM(n) define

B v A if and only if B 4 A, co(B) = co(A) and ro(B) = ro(A).

Put B @ A if B v A and B 6= A.

The order relation v

For A ∈ ΘM(n) and i 6= j ∈ Z, let

σi,j(A) =

s6i,t>jas,t, if i < j;∑

s>i,t6jas,t, if i > j.

For A,B ∈ ΘM(n), define

B 4 A if and only if σi,j(B) 6 σi,j(A) for all i 6= j.

Put B ≺ A if B 4 A and, for some pair (i, j) with i 6= j,

σi,j(B) < σi,j(A).

For A,B ∈ ΘM(n) define

B v A if and only if B 4 A, co(B) = co(A) and ro(B) = ro(A).

Put B @ A if B v A and B 6= A.

Canonical bases of affine quantum Schur algebras

Let θA,r | A ∈ ΘM(n, r) be the canonical basis of affine quantum

Schur algebras defined by Lusztig.

θA,r = θA,r, θA,r − [A] ∈∑

B∈ΘM(n,r),B@A v−1Z[v−1][B].

Canonical bases of affine quantum Schur algebras

Let θA,r | A ∈ ΘM(n, r) be the canonical basis of affine quantum

Schur algebras defined by Lusztig.

θA,r = θA,r, θA,r − [A] ∈∑

B∈ΘM(n,r),B@A v−1Z[v−1][B].

Canonical bases of modified quantum affine gln

(1) There exists a unique Z-basis θA | A ∈ ΘM(n) for KM(n)Z = DM(n)Z

such that θA = θA and θA − ·[A]· ∈∑

B∈ΘM(n),B@A v−1Z[v−1]·[B]·.

(2) There is an algebra epimorphism ζr : KM(n)Z → SM(n, r)Z such that

ζr(·[A]·) =

[A] if A ∈ ΘM(n, r);

0 otherwiseand ζr(θA) =

θA,r, if A ∈ ΘM(n, r);

0, otherwise.

(3) Let θ+A | A ∈ Θ+

M (n) be the canonical basis of D+M (n)

(Varagnolo–Vasserot). Then we have θ+A =

∑µ∈ZnM θA+diag(µ).

B∈ΘM(n),B@A v−1Z[v−1]·[B]·.

ζr(·[A]·) =

0, otherwise.

(3) Let θ+A | A ∈ Θ+

B∈ΘM(n),B@A v−1Z[v−1]·[B]·.

ζr(·[A]·) =

0, otherwise.

(3) Let θ+A | A ∈ Θ+

B∈ΘM(n),B@A v−1Z[v−1]·[B]·.

ζr(·[A]·) =

0, otherwise.

(3) Let θ+A | A ∈ Θ+

Canonical bases of UM(n)

Let UM(n) be the subalgebra of DM(n) generated by U(sln) and K±1i .

DM(n) ∼= UM(n)⊗Q(v)[z±1 , z±2 , · · · ], where z±s (s > 1) is certain

central elements of DM(n) (Schiffmann).

UM(n)(⊆ DM(n)) : the modified form of UM(n).

Call a matrix A = (ai,j) ∈ ΘM(n) to be aperiodic if for every integer

l 6= 0 there exists 1 6 i 6 n such that ai,i+l = 0. Let ΘapM (n) be the

set of all aperiodic matrices in ΘM(n).

There exists a basis θ′A | A ∈ ΘapM (n) of UM(n) such that

ζr(θ′A) = ζr(θA) =

θA,r if A ∈ ΘM(n, r);

0 if A 6∈ ΘM(n, r)for A ∈ Θap

M (n).

Remark

Lusztig conjecture [Asian J. Math., 1999] that there is a basis B of U(sln)

and an algebra homomorphism φr : U(sln)→ SM(n, r) such that

φr(B) ⊆ 0 ∪ Br, where Br := θA,r | A ∈ ΘM(n, r). This conjecture

is proved by Schiffmann–Vasserot in [Transform. Groups, 2000]. The

modified quantum affine algebra of Schiffmann–Vasserot is a homomorphic

image of UM(n).

There exists a basis θ′A | A ∈ ΘapM (n) of UM(n) such that

ζr(θ′A) = ζr(θA) =

θA,r if A ∈ ΘM(n, r);

0 if A 6∈ ΘM(n, r)for A ∈ Θap

M (n).

Remark

Lusztig conjecture [Asian J. Math., 1999] that there is a basis B of U(sln)

and an algebra homomorphism φr : U(sln)→ SM(n, r) such that

φr(B) ⊆ 0 ∪ Br, where Br := θA,r | A ∈ ΘM(n, r). This conjecture

is proved by Schiffmann–Vasserot in [Transform. Groups, 2000]. The

modified quantum affine algebra of Schiffmann–Vasserot is a homomorphic

image of UM(n).Qiang Fu (Tongji University) Quantum affine gln via Hecke algebras 52 / 53

Thank you!

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