representations of quantum toroidal algebras recent topicsrepresentations of quantum toroidal...

Representations of quantum toroidal algebras —Recent topics—

Representations of quantum toroidal algebras—Recent topics—

Michio Jimbo (Rikkyo University, Japan)Joint with Feigin, Miwa, Mukhin

Recent Advances in Quantum Integrable SystemsAngers, September 11, 2012


Brief history

• Let g be a complex simple Lie algebra.The affinization of g is a Kac-Moody Lie algebra, also realized as aone-dimensional central extension of the loop algebra g⊗ C[t±].The toroidal Lie algebra is a further affinization, realized as atwo-dimensional central extension of the double loop algebrag⊗ C[s±, t±].

• Quantum toroidal algebras are a quantization of torodial Liealgebras. They were first introduced by geometric methods in

V.Ginzburg, M.Kapranov and E.Vasserot,Math. Res. Lett. 2 (1995) 147–160

The case g = gln is special, in that one can introduce twodeformation parameters in the algebra. From now on we consideronly the gln case.


• Some early studies include the Schur-Weyl duality

M.Varagnolo and E.Vasserot, Commun. Math. Phys.182 (1996) 469–484

and the construciton of Fock modules

Y.Saito, K.Takemura and D.Uglov, Transf. Groups 1(1998) 75–102

• Basic theory on the structure and representations has beenestablished by a series of works by Miki (algebraic method a laChari-Pressley and Beck)

K. Miki, Lett. Math. Phys. 40 (1999) 365–378;J. Math. Phys. 41 (2000) 7079–7098, 42 (2001)2293–2308, 48 (2007) 123520;


• Among others, the gl1 toroidal algebra is particularly interestingfor its simplicity and extra symmetry.Originally introduced by Miki, also rediscovered by several authors,in connection with:Macdonald functions and Virasoro/W-algebras,

B.Feigin, K.Hashizume, A.Hoshino, J.Shiraishi andS.Yanagida, J. Math. Phys. 50 (2009) 095215,B.Feigin and A. Tsymbaliuk, arXiv:0904.1679, Kyoto J.Math. 51 (2011) 831–854,

double affine Hecke algebra (DAHA),

O.Schiffmann, arXiv:1004.2575 ,

and others.


Very little has been known about concrete representations ofquantum toroidal algebras (the only example: Fockrepresentation). Our goal here is to explain an elementary/explicitconstruction of a large family of modules.

This talk is based on papers by B.Feigin, T.Miwa, E.Mukhin andMJ:

Quantum toroidal gl1 algebra : plane partitions,arXiv:1110.5310, to appear in Kyoto J. Math.

Representations of quantum toroidal glN , arXiv:1204.5378,

and earlier ones by B.Feigin, E.Feigin, T.Miwa, E.Mukhin and MJ:

Kyoto J. Math. 51 (2011) 337–364; ibid. 365–392


Plan of this talk

We shall mainly focus on the gl1 case.

1 Basic definitions and known facts

2 Macmahon modules

3 Construction via Fock modules

4 Macmahon modules with boundary conditions

5 Resonances

6 gln case


Quantum gl1 toroidal algebra E

The algebra E is defined from the following data.

parameters: q1, q2, q3 ∈ C×, q1q2q3 = 1generators: Ek , Fk , Hr , K±0 , C 1/2 (k ∈ Z, r ∈ Z\{0})We set

E (z) =∑k∈Z

Ekz−k , F (z) =∑k∈Z

Fkz−k ,

K±(z) = K±0 exp(±(q

1/22 − q

−1/22 )

∑±r>0

Hrz−r),

g(z ,w) = (z − q1w)(z − q2w)(z − q3w) ,

g(z ,w) = (q1z − w)(q2z − w)(q3z − w) .


Relations:

K±0 ,C 1/2: central invertible,

K±(z)K±(w) = K±(w)K±(z),

g(Cz ,w)g(z ,Cw)K+(z)K−(w) = g(z ,Cw)g(Cz ,w)K−(w)K+(z) ,

g(z ,w)K±(C∓1/2z)E (w) = g(z ,w)E (w)K±(C∓1/2z),

g(z ,w)K±(C±1/2z)F (w) = g(z ,w)F (w)K±(C±1/2z),

[E (z),F (w)] =1

g(1, 1)(δ(Cw/z)K+(C 1/2w)− δ(Cz/w)K−(C 1/2z)),

g(z ,w)E (z)E (w) = g(z ,w)E (w)E (z),

g(z ,w)F (z)F (w) = g(z ,w)F (w)F (z),

[E0, [E1,E−1]] = [F0, [F1,F−1]] = 0.


Remark

1 Various names:

(q, γ)-analog of W1+∞ (Miki),Ding-Iohara algebra (Feigin, Shiraishi,· · · ),elliptic Hall algebra (Schiffmann)

2 E is symmetric in q1, q2, q3 (not true for En, n ≥ 2)

3 E is isomorphic to the spherical DAHA (Schiffmann 2010)


SL(2,Z)-action

Automorphisim [Miki 2007]

There exists an order 4 autmorphism ψ of E such that

H1 7→ c1E0, H−1 7→ c2F0,

E0 7→ −c−11 H−1, F0 7→ −c−12 H1,

C 7→ K0, K0 7→ C−1


• Hereafter we assume that qi ’s are generic, i.e.,

qj11 qj2

2 qj33 = 1 only if j1 = j2 = j3.

• An E-module V is of level (x , y) if C acts as sclar x and(K+

0 )−1K−0 as y .

• In the rest of this talk we consider modules of level (1, y), i.e.C = 1. Then the operators K±(z) are mutually commutative.


Lowest weight modules

An E-module V is called lowest weight if V = Ev with

F (z)v = 0 , K±(z)v = φ±(z)v , Cv = v ,

for some formal power series φ±(z) in z∓1. (φ+(z), φ−(z)) iscalled the lowest weight of V .Assign degrees by

deg Ek = 1, deg Fk = −1, deg Hr = 0 .

A graded E-module V =⊕k∈Z

Vk is called quasi-finite if

dim Vk <∞ (∀k).


Classification [Miki 2007]

1) An irreducible lowest weight module M is in one-to-onecorrespondence with its lowest weight (φ+(z), φ−(z)).2) M is quasi-finite if and only if φ±(z) are expansions of acommon rational function φ(z) in z∓1.

So, given a rational function φ(z) which is regular and non-zero atz = 0,∞, there corresponds a unique irreducible lowest weightE-module M. From the definition, however, it is hard to tell how itlooks like. For instance, what is the character

χ(M) =∑k

qk dim Mk .


Macmahon modules

A Macmahon module M(u,K ) is the irreducible lowest weightmodule whose lowest weight is

φ(z) =1− Ku/z

1− u/z(u,K ∈ C).

We shall construct M(u,K ) explicitly.

Main features:

1 M(u,K ) has a basis parametrized by all plane partitions,

2 K±(z) act diagonally on them with simple joint spectrum,

3 E (z) adds a box and F (z) removes a box, with explicitly givenmatrix coefficients.


A plane parition λ = (λ(1), λ(2), λ(3), · · · ) is a sequence of ordinarypartitions (Young diagrams) satisfying

λ(1) ⊃ λ(2) ⊃ λ(3) ⊃ · · · , λ(N) = ∅ (N � 0).

Denote the set of all plane partitions by π.We visualize λ ∈ π as a 3-dimensional Young diagram Yλ, wherethe k-th Young daigram λ(k) is “sitting on the k-th floor”.


3D Young diagram Yλ

λ = (λ(1), λ(2), λ(3))

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Let M(u,K ) be a vector space with basis |λ〉 (λ ∈ π).There is an action of E on M(u,K ) of the form

K±(z)|λ〉 = ψλ(u/z)|λ〉,

(1− q1)E (z)|λ〉 =∑

(i , j , k) : concave

ψλ,i ,j ,kψλ(k),iδ(qi3qj

1qk2

)|λ+ 1

(k)i 〉 ,

(q−11 − 1)F (z)|λ〉 =∑

(i , j , k) : convex

ψ′λ,i ,j ,kψ′λ(k),i

δ(qi3qj

1qk2

)|λ− 1

(k)i 〉 ,

All coefficients have factorized form, e.g.,

ψλ(u/z) =1− Ku/z

1− u/z

∏(i ,j ,k)∈Yλ

h(qi3qj

1qk2u/z) ,

h(z) =(1− q−11 z)(1− q−12 z)(1− q−13 z)

(1− q1z)(1− q2z)(1− q3z).


For generic K , M(u,K ) is an irreducible, tame, lowest weight

E-module of lowest weight1− Ku/z

1− u/zand level (1,K ).

By the Macmahon formula for plane partitions, we have

χ(M(u,K )

)=∞∏i=1

1

(1− qi )i.


Idea of construction

Fock module for gl∞ is constructed as a semi-infinite wedge

Fgl∞ = V ∧ V ∧ V ∧ · · · , V = C∞

We construct the Macmahon module analogously, followingtwo-step semi-infinite constructions

V (u) =⇒ F(u) =⇒ M(u,K )

The main points are the Drinfeld coproduct and the structures ofzeroes of the matrix coefficients.


Vector representation

Let V (u) =⊕i∈Z

C[u]i .

The following formula defines an E-module structure on V (u):

(1− q1)E (z)[u]i = δ(qi1u/z

)[u]i+1 ,

(q−11 − 1)F (z)[u]i = δ(qi−11 u/z

)[u]i−1 ,

K±(z)[u]i = ψ(qi1u/z

)[u]i ,

where

ψ(z)

=(1− q2z)(1− q3z)

(1− z)(1− q2q3z).


Drinfled coproduct

Algebra E has the following formal coproduct (C = 1, forsimplicity)

∆E (z) = E (z)⊗ 1 + K−(z)⊗ E (z),

∆F (z) = F (z)⊗ K+(z) + 1⊗ F (z),

∆K±(z) = K±(z)⊗ K±(z),

The right hand side is an infinite series, so it is not defined in theusual sense.


Example. In V (u)⊗ V (v),

(1− q1)E (z)[u]i ⊗ [v ]j = δ(qi1u/z)[u]i+1 ⊗ [v ]j

+1− q2qi−j

1 u/v

1− qi−j1 u/v

1− q3qi−j1 u/v

1− q2q3qi−j1 u/v

× δ(qj1v/z)[u]i ⊗ [v ]j+1

For generic u, v , V (u)⊗ V (v) is an well-defined irreducibleE-module. (For special values of u/v , poles can appear.)

Take v = q−12 u. Then

Poles do not occur,

The subspace

span{[u]i ⊗ [uq−12 ]j | i > j} ⊂ V (u)⊗ V (q−12 u)

is invariant under E (z) because of the zero1− q3qi−j

1 u/v = 0 .

Easy to check that this subspace is a well-defined E-module. (Itcan be thought of as an analog of ∧2V (u).)


Example. In V (u)⊗ V (v),

(1− q1)E (z)[u]i ⊗ [v ]j = δ(qi1u/z)[u]i+1 ⊗ [v ]j

+1− q2qi−j

1 u/v

1− qi−j1 u/v

1− q3qi−j1 u/v

1− q2q3qi−j1 u/v

× δ(qj1v/z)[u]i ⊗ [v ]j+1

For generic u, v , V (u)⊗ V (v) is an well-defined irreducibleE-module. (For special values of u/v , poles can appear.)Take v = q−12 u. Then

Poles do not occur,

The subspace

span{[u]i ⊗ [uq−12 ]j | i > j} ⊂ V (u)⊗ V (q−12 u)

is invariant under E (z) because of the zero1− q3qi−j

1 u/v = 0 .

Easy to check that this subspace is a well-defined E-module. (Itcan be thought of as an analog of ∧2V (u).)


A semi-infinite construction

Consider the subspace of the infinite tensor product

F(u) ⊂ V (u)⊗ V (uq−12 )⊗ V (uq−22 )⊗ · · ·F(u) = span{|λ〉 | λ1 ≥ λ2 ≥ · · · , λi ∈ Z , λi = 0 (i � 0)}

where|λ〉 = [u]λ1 ⊗ [uq−12 ]λ2−1 ⊗ [uq−22 ]λ3−2 ⊗ · · · .

The action of E (z) is well defined on F(u).

The actions of F (z), K±(z) need interpretation as they involve aformal infinite product. We define a modified action as follows.


A semi-infinite construction

Consider the subspace of the infinite tensor product

F(u) ⊂ V (u)⊗ V (uq−12 )⊗ V (uq−22 )⊗ · · ·F(u) = span{|λ〉 | λ1 ≥ λ2 ≥ · · · , λi ∈ Z , λi = 0 (i � 0)}

where|λ〉 = [u]λ1 ⊗ [uq−12 ]λ2−1 ⊗ [uq−22 ]λ3−2 ⊗ · · · .

The action of E (z) is well defined on F(u).The actions of F (z), K±(z) need interpretation as they involve aformal infinite product. We define a modified action as follows.


Given a partition λ, take N large enough and consider a vector

|λ〉(N) = [u]λ1 ⊗ [uq−12 ]λ2−1 ⊗ · · · ⊗ [uq−N+22 ]−N+2 ⊗ [uq−N+1

2 ]−N+1

in the finite tensor product

F (N)(u) = V (u)⊗ V (uq−12 )⊗ V (uq−22 )⊗ · · · ⊗ V (uq−N+12 ) .

Then the modified action

F new (z)|λ〉(N) = βN(z) · F (z)|λ〉(N) ,

K±,new (z)|λ〉(N) = βN(z) · K±(z)|λ〉(N) ,

βN(z) =1− qN

3 u/z

1− q−12 qN3 u/z

is compatible with the embedding F (N)(u) ↪→ F (N+1)(u).


Fock module

With the above definition, F(u) is a well-defined E-module.F(u) is an irreducible, tame, lowest weight module of lowest

weight1− q2u/z

1− u/zand level (1, q2).

All matrix coefficients are factorized. e.g.

〈λ|K±(z)|λ〉 =∏

(i , j):concave

1− qi3qj

1q22u/z

1− qi3qj

1q2u/z

∏(i , j):convex

1− qi3qj

1u/z

1− qi3qj

1q2u/z,

In this form, F(u) was found by Feigin-Tsymbaliuk with ageometric method.


From Fock to Macmahon

One can repeat the same construction, using F(u) in place ofV (u). We use

Lemma: Let a, b ∈ Z≥0, and let

v/u = q−a1 q2q−b3 .

Then the subspace

span{|λ〉 ⊗ |µ〉 | λi + a ≥ µi+b (∀i)} ⊂ F(u)⊗F(v)

is an E-submodule.


Choosing a = b = 0 in the Lemma, consider the subspace of aninfinite tensor product of Fock modules

M(u,K ) ⊂ F(u)⊗F(uq2)⊗F(uq22)⊗ · · ·

M(u,K ) = span{|λ〉 | λ ∈ π}

where for λ = (λ(1), λ(2), · · · ) we set

|λ〉 = |λ(1)〉 ⊗ |λ(2)〉 ⊗ · · · ⊗ |∅〉 ⊗ · · · .

As before, E (z) has a well-defined action on M(u,K ).For each λ ∈ π, the modification

F new (z)|λ〉(N) =1− Ku/z

1− u/z· F (z)|λ〉(N) ,

K±,new (z)|λ〉(N) =1− Ku/z

1− u/z· K±(z)|λ〉(N) ,

leads to the action of F (z),K (z) on M(u,K ).


Macmahon modules with boundary conditions

More generally, let α, β, γ be given partitions. Let Mα,β,γ(u,K ) bethe vector space spanned by infinite plane partitions µ such that

µ(k)i = αk (i � 0) ,

µ(k)i = γi (k � 0) ,

µ(k)i =∞ if 1 ≤ i ≤ βk .

Let ω be the minimal plane partition with this boundary condition.

There is an action of E on Mα,β,γ(u,K ). It is an irreducible,tame, lowest weight module of level (1,K ) and lowest vector|ω〉.


Macmahon module with boundary (α, β, γ).

The minimal diagram Yω is shown.

γ

α β

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Lowest weights of Mα,β,γ(u,K )

α β γ Lowest weight

∅ ∅ ∅ 1− Ku/z

1− u/z

{1} ∅ ∅ (1− Ku/z)(1− q1q2u/z)

(1− q1u/z)(1− q2u/z)

{2} ∅ ∅(1− Ku/z)

(1− q2

1q2u/z)(

1− q21u/z

)(1− q2u/z)

{1} {1} ∅ (1− Ku/z)(1− q1q2q3u/z)

(1− q2u/z)(1− q1q3u/z)

{1} {1} {1} (1− Ku/z)(1− q1q2q3u/z)2

(1− q1q2u/z)(1− q1q3u/z)(1− q2q3u/z)


Resonances

When the parameters K , u or q1, q2, q3 take special values,Mα,β,γ(u,K ) become reducible. We call this a resonant case.

We consider only the case where q1, q2, q3 are generic but K isspecial:

K = qm2 qn

3 (m, n ∈ Z≥0) ,

In this case, the action of F (z) cannot remove the boxes at(a + t, b + t, c + t) (y ∈ Z≥0), where

(a, b, c) ∈ Yω, (a− 1, b − 1, c − 1) 6∈ Yω.

Correspondingly there is a series of submodules

Mα,β,γ(u,K ) = Mm,n,0α,β,γ (u) ⊃ Mm,n,1

α,β,γ (u) ⊃ Mm,n,2α,β,γ (u) ⊃ · · ·


Let

Nm,nα,β,γ(u) := Mm,n,0

α,β,γ (u)/Mm,n,1α,β,γ (u)

be the first irreducible quotient.

For example, if K = q2 then M1,0,1∅,∅,∅ (u) contain all diagrams

containing the box (0, 0, 1). Hence

N1,0∅,∅,∅(u) = F(u) .


Characters of Nn,nα,∅,∅(u)

In general, it is a challenge to determine the character ofNm,nα,β,γ(u). The following result is a special case.

χk = χk , χ−k = qk χk (k ∈ Z≥0) ,

χk =1

(q)2∞

∞∑j=0

(−1)jqj(j+1)/2+jk .

χ(Nn,nα,∅,∅

)=∑σ∈Sn

(−1)`(σ)n∏

i=1

χ(σ(α+ρ)−ρ)i .


gln case (n ≥ 3)

Almost all constructions for the gl1-quantum toroidal algebra carryover to the case of gln. Minor differences are:

Algebra is symmetric in q3, q1 but q2 plays a different role.,

Boxes in (plane) partitions are colored ((i , j , k) has colori − j mod n),

Macmahon modules with boundary conditions can be definedonly when partition γ is colorless.


Summary

1 We have obtained a new family of modules of quantumtoroidal algebras.

2 The actions of generators are explicit and the joint spectrumof the K (z)’s is simple.

3 This allows a combinatorial study of these modules.

Open questions

1 Characters

2 Resonances in K (e.g. K = q1q2q3)

3 Resonances in q1, q2, q3

4 Are there integrable models which have E as its symmetry?


THANK YOU VERY MUCH FOR YOUR ATTENTION

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