Integrable modules for twisted toroidal extended affine Lie algebras

arXiv:1802.09823v1 [math.RT] 27 Feb 2018

INTEGRABLE MODULES FOR TWISTED TOROIDAL EXTENDED AFFINE LIE ALGEBRAS S. ESWARA RAO, SACHIN S. SHARMA AND PUNITA BATRA

Abstract. In this paper we classify the irreducible integrable modules for the twisted toroidal extended affine Lie algebras with center acting non-trivially.

MSC: 17B67,17B69 KEY WORDS: Divergence zero vector fields, Lie torus, Extended affine Lie algebras.

1. Introduction In order to unravel the secret behind success of representation theory of Kac-Moody algebras, one tries to generalize them. Extended affine Lie algebras (EALAs, for short) are natural generalization of Kac-Moody algebras; finite dimensional simple Lie algebra, affine Kac-Moody algebras, toroidal Lie algebras are examples of EALAs. They first appeared in the work of Saito and Slodowy on elliptic singularities and in the paper of physicists Høegh and Torresani in 1990. Later mathematicians like Allison, Azam, Berman, Gao, Pianzola, Neher and Yoshii systematically developed theory of EALAs (see [13, 15, 1], and references therein). Unlike Kac-Moody algebras, EALAs may have infinite dimensional centers. So representation theory of EALA’s is still in progress. 1

2

S. ESWARA RAO, SACHIN S. SHARMA AND PUNITA BATRA ◦

◦

◦

Let L(g) = g ⊗ An is a loop algebra, where g is a finite dimensional simple Lie algebra and An is a commutative Laurent polynomial ring of n variables. ◦

◦

Let L(g)⊕ΩAn /dAn be universal central extension of L(g), and let Sn be the Lie algebra of divergence zero vector fields (see Section 3 for more details). ◦

Then the Lie algebra L(g) ⊕ ΩAn /dAn ⊕ Sn is called toroidal EALA. In [3] Billig constructed irreducible representation of toroidal EALAs using vertex operator algebras. In [4] Billig and Lau constructed irreducible modules for twisted toroidal EALAs. In this paper we classify irreducible integrable modules for twisted toroidal EALAs where center acts non-trivially. In section 3 we start with Lie algebra of derivations Der(An ) which we denote by D and consider the subalgebra T of U (D ⋊ An ). We consider the ideals Ik of T and prove that their intersection is zero (Proposition 3.2). Using this we prove that every Sn ⋊An irreducible module with finite dimensional weight spaces and associative action of An is isomorphic to F α,1 (V ), where V is a finite dimensional irreducible module for sln , which improves the result in [14]. Our result is also finer than result in [5] (Theorem 5.2) as it follows that the abelian Lie algebra an must act trivially on irreducible ◦

modules. In section 4 for the loop algebra L(g) we define ideals Fd similarly as in [2] and and prove that intersection of ideals Fd is zero (Proposition 4.1). In section 5 we define our main object of study twisted toroidal EALAs T , by defining full toroidal Lie algebra and toroidal EALAs. We start with an irreducible integrable module V with finite dimensional weight spaces of T with condition that the central elements K0 , K1 , . . . , Kn act nontrivially. Up to an automorphism we can assume that K0 acts as C0 ∈ Z>0 and Ki acts trivially for 1 ≤ i ≤ n. Rest of the paper follows in the similar path as

INTEGRABLE MODULES FOR TWISTED TOROIDAL EXTENDED AFFINE LIE ALGEBRAS 3

in [2]. We define T = T − ⊕ T 0 ⊕ T + a natural triangular decomposition of T and prove that V is highest weight module with respect to it and its the highest weight space M is a nonzero T 0 irreducible module. Then we use result in [10] to discard the elements of T 0 which act as scalars and consider its subalgebra L. Then L and M are Zn graded and M is Zn graded module of L. As like in [2] we consider the subalgebra I of L, which is ungraded and its finite dimensional module V˜ which is quotient of M . Then by [2] (Theorem 8.3) it follows that V˜ is completely reducible I module with isomorphic irreducible components. So we consider the finite dimensional irreducible module of I and classify them. It turns out to be an irreducible module for sln ⊕ g(0). Here we would like to point out that Proposition 3.2 plays a crucial role in this classification result as techniques in [2] won’t work in the present case. We lift I module V˜ to L-module L(V˜ ) similarly as in [2]. In case V˜ is irreducible I module then M can be identified with the one of the irreducible components of L(V˜ ), while in case of reducibility, M is still submodule of L(V˜ ) with inclusion being twisted. Finally, we start with sln ⊕ g(0)-module and give it L structure, which contains an isomorphic copy of M a T 0 irreducible module, and obtain our irreducible integrable module V for T as a quotient of an induced module of M .

2. Notations Let C, R, denote the fields of complex numbers and real numbers. Let N be set natural numbers, let Z be set of integers and Z≥0 , Z>0 denote the set nonnegative and positive integers respectively. We denote Zn≥0 = {(r1 , . . . , rn ) ∈

4

S. ESWARA RAO, SACHIN S. SHARMA AND PUNITA BATRA

Zn |ri ∈ Z≥0 , 1 ≤ i ≤ n} and Zn>0 = {(r1 , . . . , rn ) ∈ Zn |ri ∈ Z>0 , 1 ≤ i ≤ n}. Let (·|·) denote the standard inner product on Cn . For k = (k1 , . . . , kn ) ∈ Zn , tk = tk11 · · · tknn ∈ An . Let for r = (r1 , . . . , rn ) ∈ Zn≥0 , |r| := r1 + · · · + rn and

d|r| dt

:=

dr1 dt1

rn

· · · ddtn . For any Lie algebra L, U (L) denotes its universal

enveloping algebra.

3. ±1 In this section we improve Xu’s [14] result. Let An = C[t±1 1 , . . . , tn ] be

a Laurent polynomial ring in n variables, where n ≥ 2. Let D = Der(A) be the Lie algebra of derivations on An . Recall that {di , tr di | 0 6= r ∈ Zn , i = 1, . . . , n} is a basis for D, where di = ti dtdi . Let for u ∈ Cn and r ∈ Zn , define Pn r D(u, r) = i=1 ui t di . We have [D(u, r), D(v, s)] = D(w, r + s), where w = (u|s)v −(v|r)u. Also recall the element T (u, r) = t−r D(u, r)−D(u, 0) ∈

U (M ⋊ A) and T = span{T (u, r)|u ∈ Cn , r ∈ Zn }. It follows that T is a Lie algebra by the following identity:

[T (u, r), T (v, s)] = (v|r)T (u, r) − (u|s)T (v, s) + T (w, r + s).

Let for k ∈ N, u ∈ Cn , r, m1 , . . . , mk ∈ Zn , Tk (u, r, m1 , m2 , . . . , mk ) = P P T (u, r) − ni=1 T (u, r + mi ) + i 0, then λ − γ ∈ P (V ). (5) For λ ∈ P (V ), λ(Ki ) is an integer independent on λ.

6.1.

Now rest of this paper we will work with irreducible integrable rep-

resentation V of T with central element K0 acts as C0 ∈ Z>0 and rest of Ki ’s act trivially for 1 ≤ i ≤ n. We now define an order on H∗ . Let for ◦

◦

λ ∈ H∗ , λ′ denote the restriction of λ to h(0)∗ and for a given λ′ ∈ h(0)∗ , extend it to H∗ by defining λ′ (Ki ) = 0 and λ′ (di ) = 0 for 0 ≤ i ≤ n. Then n n X X λ(di )δi and for λ(Ki )ωi + we have an unique expression for λ = λ′ + i=0

λ ∈ P (V ) then λ = λ′ + λ(K0 )ω0 + λ(d0 )δ0 + ¯+ we write λ = λ

n X

n X

i=0

λ(di )δi . So for λ ∈ P (V ),

i=i

¯ = λ′ + λ(K0 )ω0 + λ(d0 )δ0 . Let β0 λ(di )δi , where λ

i=i

be a maximal root in ∆0,en . Define α0 = −β0 + δ0 , which may not be root ◦

◦

T . Let α1 , . . . , αq be simple roots of ∆(g(¯0, ¯0), h(0)) and define the lattice q M Nαi . Let for Λ, λ ∈ H∗ , we define an order on H by λ ≤ Λ if Q+ = i=0

Λ − λ ∈ Q+ . It is easy to see that for Λ ≤ λ, Λ(di ) = λ(di ) for 1 ≤ i ≤ n. Consider the natural triangular decomposition of T :

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M

LT + =

¯ α) ⊗ tk0 tk ; g(k¯0 , k, 0

α+k0 δ0 >0, k∈Zn

M

LT − =

¯ α) ⊗ tk0 tk ; g(k¯0 , k, 0

α+k0 δ0 0}; − Sn+1 (m0 , m) ={D(u, r) | u ∈ Cn+1 , r ∈ Γ0 ⊕ Γ, (u|r) = 0, r0 < 0}; 0 Sn+1 (m0 , m) ={D(u, r) | u ∈ Cn+1 , r ∈ Γ0 ⊕ Γ, (u|r) = 0, r0 = 0};

Z+ =

M

Cts00 ts Ki ;

M

Cts00 ts Ki ;

0≤i≤n 0s0 ∈Γ0 ,s∈Γ

Z0 =

M

Cts Ki .

0≤i≤n,s∈Γ + − Let T + = LT + ⊕ Z + ⊕ Sn+1 (m0 , m), T − = LT − ⊕ Z − ⊕ Sn+1 (m0 , m) and 0 (m , m). Then T = T − ⊕ T 0 ⊕ T + is a trigular T 0 = LT 0 ⊕ Z 0 ⊕ Sn+1 0

decomposition of T . We have the following Theorem 6.3. Let M = {v ∈ V | T + v = 0} = 6 0. Proof. It will suffice to show the existence of γ ∈ P (V ) such that γ +β +δk ∈ / P (V ) for any root β + δk ∈ P (V ) with β > 0 and k ∈ Zn . This is similar to Theorem 5.2 of [2].

It follows that M is a irreducible T 0 -module and U (T − )M = V . Note 0 (m , m). Let γ ∈ P (V ) from the Theorem that M is Zn -graded as di ∈ Sn+1 0


6.3. Let Mr = {v ∈ M |di v = (γ(di ) + ri )v, 1 ≤ i ≤ n}. We will need the following results which can be proved similarly as proved in [10]. (1) ts K0 v 6= 0, for any nonzero v ∈ M and ∀s ∈ Γ

Proposition 6.4.

and dimMk = dimMk+s = zk , ∀s ∈ Γ. (2) If {v1 , . . . , vzk } is a basis of Mk then {v1 (k + s), . . . , vzk (k + s)} is a basis for Mk+s , where

1 s C0 t K0 vi (k)

= v(k + s) and

1 s C0 t K0 (v1 (k

+

r), . . . , vzk (k + r)) = (v1 (k + r + s), . . . , vzk (k + r + s)). ◦

(3) For h ∈ h(0) and γ be fixed element of P (V ), h⊗ts (v1 (k+r), . . . , vzk (k+ r)) = γ(h)(v1 (k+r+s), . . . , vzk (k+r+s)) and ts d0 (v1 (k+r), . . . , vzk (k+ r)) = γ(d0 )(v1 (k + r), . . . , vzk (k + r)). (4) tr Kd .M = 0 ∀ 1 ≤ d ≤ n, r ∈ Γ and tr K0 ts K0 v = C0 tr+s K0 , ∀ v ∈ M, r, s ∈ Γ. By above proposition it follows that M ∼ = denote V 1 =

M

M

0≤ki ≤mi 0≤i≤n

Mk

!

⊗ A(m). Let

Mk . As like in [2] S(n+1) (m0 , m)0 is identified with

0≤ki ≤mi

X 0≤i≤n X Sn (m) ⊕ Ctr d0 , and Z 0 with Ctr K0 , since the rest of the space acts r∈Γ

r∈Γ

trivially on M . Now using (4) of above proposition we identifying tr K0 with tr for r ∈ Γ. Hence V 1 ⊗ A(m) becomes an irreducible module for the space X L = LT 0 ⊕ Sn (m) ⊕ Ctr d0 ⊕ A(m). Now we will rewrite LT 0 for our r∈Γ

◦

◦

convenience: for that let g(0) = {X ∈ g | σ0 (X) = X, [h, X] = 0, h ∈ h(0)}. ◦

Then g(0) 6= 0 as h(0) ⊆ g(0) and σi (g(0)) ⊆ g(0). So g(0) preserves ΛM M grading and let g(0) = (g(0))k¯ and let L g(0), σ = (g(0))k¯ ⊗ tk ¯ k∈Λ

k∈Z n+1

be the corresponding multiloop algebra. Then it is easy to see that LT 0 = L g(0), σ .

20


Again as in [2] W be the subspace spanned by ts v(p) − v(p), where v(p) = v⊗tp , v ∈ V 1 and s, p ∈ Γ. Then W is module for S ′ (Γ)⋊L g(0), σ ⊕A(m)⊗ X Ctr d0 , where S ′ (Γ) = span{D(u, r) − D(u, 0) : u ∈ Cn , r ∈ Γ, (u|r) = 0}. r∈Γ

It is easy to see that S ′ (Γ) is a Lie algebra and isomorphic (as Lie algebras)

to Φ(T ′ ) = S ′ . Let us consider Ve = (V 1 ⊗ A(m))/W . By previous argument X X Ve is an S ′ (Γ) ⋊ L g(0), σ ⊕ A(m) ⊕ Ctr d0 . As A(m) ⊕ Ctr d0 act as

r∈Γ r∈Γ scalars on Ve , we neglect them and consider it as I := S ′ (Γ) ⋊ L g(0), σ -

module. Now define L(V ) = Ve ⊗ An , where V is any I module. Then L(Ve ) is a L-module by the following action:

X(k).v ⊗ ts = (X(k)v) ⊗ tk+s ; D(u, r).(v ⊗ ts ) = (S(u, r)v) ⊗ tr+s + (u|s + µ)v ⊗ ts+r ; tl .v ⊗ ts = v ⊗ ts+l ; tl d0 .v ⊗ ts = γ(d0 ).v ⊗ tt+s , where v ∈ V, s, l ∈ Γ, u ∈ Cn , r ∈ Γ with (u|r) = 0. Theorem 6.5. Ve is completely reducible as I module with isomorphic irreducible components.

Proof. Proof of this theorem is exactly similar to Theorem 8.3 of [2].

Remark 6.6. Using Λ-gradation on Ve one can define Λ-gradation on L(Ve ). In case if Ve is irreducible as I-module it follows that M is isomorphic as

T 0 -module to the zeroth graded piece of L(e(V )) with respect to Λ-gradation. When Ve is reducible as I-module, inclusion of M in L(Ve ) is more complex. See Section 8 of [2] for more details.


Now using above theorem, let W be an irreducible finite dimensional module of I, and let ρ : I 7→ End(W ) be the corresponding representation. Let K be the kernel of ρ. Then it is easy to see that K is a cofinite ideal of L(g(0), σ) and hence contains Fd for some large d by Proposition 4.1. Now as [Fd−1 , Fd−1 ] ⊆ Fd , it follows from Lemma 9.2 of [2] that ρ(Fd−1 ) is central ideal in Φ(I) and hence acts as a scalar. Then by similar argument as Lemma 9.5 in [2] , this scalar has to be zero and hence Fd−1 ⊆ K, and inductively we have F1 ⊆ K. Now as [S ′ (Γ), L(g(0), σ)] ⊆ F1 , it follows that W is an irreducible module for the Lie algebra S(Γ) ⊕ L(g(0), σ). Now we need the following result due to H. Li :

Proposition 6.7 (Lemma 2.7, [11]). Let A1 and A2 be associative algebras with identity. Let U be an irreducible A1 ⊗ A2 - module. Suppose that U as A1 -module has an irreducible submodule U1 and assume that either A1 is countable dimension or EndA1 U1 = C. Then U ∼ = U1 ⊗ U2 as A1 ⊗ A2 module, where U2 is an irreducible U2 -module.

Taking U1 = U (S ′ (Γ)) and U2 = U (L(g(0), σ)), we see that hypothesis of above proposition is satisfied as W is finite dimensional and U (S) has countable dimension over C. we see that W ∼ = W1 ⊗ W2 , where W1 and W2 are finite dimensional irreducible modules for S ′ (Γ) and L(g(0), σ). But as L(g(0), σ)/F1 ∼ = g(0) (Lemma 7.1, [2]) and S ′ (Γ) ∼ = T ′ , by Remark 3.6 = S′ ∼ it follows that W1 ⊗ W2 is an irreducible module for sln ⊕ g(0).

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7. Description of modules for T Recall that T 0 = LT 0 ⊕ Z 0 ⊕ S 0n+1 (m0 , m), where Z 0 = and

M

Cts Ki

0≤i≤n,s∈Γ X r is identified with Sn (m) ⊕ Ct d0 . Now let, W1 and

0 (m , m) Sn+1 0

r∈Γ

W2 be irreducible modules for sln and g(0) respectively with W2 being Λgraded which is compatible with g(0). Consider the space W1 ⊗ W2 ⊗ An . Pn We will define T 0 action on it. But first let u ∈ Cn , r = i=1 ri mi ei

such that (u|r) = 0. As seen before let Iu,r be the set indices i such that

ui ri mi 6= 0, and let Iu,r = {i1 , . . . , iq }, where i1 < i2 < · · · < iq . Now for α ∈ Cn , l, k ∈ Zn , s ∈ Γ, X ∈ g(0), w1 ∈ W1 and w2 ∈ W2 , define

X(k)(w1 ⊗ w2 ⊗ tl ) = w1 ⊗ (X(w2 ) ⊗ tl+k ); l

D(u, r)(w1 ⊗w2 ⊗t ) =

q−1 X a=1

+

X i6=j

ia X

!

!

ub rb mb (Eia ,ia − Eia+1 ,ia+1 )w1 ⊗w2 ⊗tr+l

b=1

ui rj mj Ej,i w1 ⊗ w2 ⊗ tl+k + (u|l + α)w1 ⊗ w2 ⊗ tr+l ;

ts d0 (w1 ⊗ w2 ⊗ tl ) = γ(d0 )(w1 ⊗ w2 ⊗ ts+l ); 1 s C0 t K0 (w1

⊗ w2 ⊗ tl ) = w1 ⊗ w2 ⊗ ts+l ;

ts Kd (w1 ⊗ w2 ⊗ tl ) = 0, 1 ≤ d ≤ n. 0 (m , m). It also One can check that W1 ⊗ W2 ⊗ An is g(0)⊗ An ⊕ Z 0 ⊕ Sn+1 0 M T 0 - module structure by considering Λ gradation of W2 = W2,k¯ which ¯ k∈Λ

is compatible with Λ-gradation of g(0). Then it follows that the submodule M M′ = W1 ⊗ W2,k¯ ⊗ tk is an irreducible module T 0 . Consider the induced k∈Zn

module M = IndT 0 ⊕T + M ′ , with T + acting trivially on M ′ . Let Mrad be the

unique maximal submodule of M. Then M/Mrad is an ireducible module for T . We have the following:


Theorem 7.1. Let V be an irreducible integrable T module with finite dimensional weight spaces, with K0 acting as C0 ∈ Z>0 and rest of Ki ’s act trivially for 1 ≤ i ≤ n. Then V ∼ = M/Mrad .

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School of mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India. email: [email protected] Department of mathematics and statistics, IIT Kanpur, Kalyanpur, Kanpur, 208016, India. [email protected] Harish-Chandra Research Institute, HBNI, Allahabad 211019, India. [email protected]