Integrable modules for twisted toroidal extended affine Lie algebras

Abstract

In this paper we classify the irreducible integrable modules for the twisted toroidal extended affine Lie algebras (twisted toroidal EALA, in short) with finite dimensional weight spaces when the finite dimensional center acts non-trivially. Using an automorphism we reduce to the case where K0 acts non-trivially and Ki, 1 ≤ i ≤ n act trivially. Twisted toroidal EALA has natural triangular decomposition and we prove that any irreducible integrable module of it with finite dimensional weight spaces is a highest weight module with respect to the above triangular decomposition. The highest weight space is an irreducible module for the zeroth component of the twisted toroidal EALA.We then describe the highest weight space in detail.MSC: 17B67,17B66 systematically developed the theory of EALAs (see [1,21,24], and references therein). Unlike affine Kac-Moody algebras, EALAs may have infinite dimensional centers. So representation theory of EALA's is still in progress.Let L ( • INTEGRABLE MODULES FOR TWISTED TOROIDAL EXTENDED AFFINE LIE ALGEBRAS 3 finite order automorphisms σ 0 , . . . , σ n of • g and consider the multi-loop alge-We assume that multi-loop algebra is a Lie torus (see Section 2 for definition). This is not a very serious assumption as it is well known that centerless cores of almost all EALAs are Lie tori [1].We then consider universal central extension of multi-loop algebra and add S n+1 the Lie algebra consisting of divergence zero vector fields on A n+1 (see Section 2 for details). The resulting Lie algebra is EALA called as twised toroidal EALA, and we denote it by T . The aim of this paper is to classify all irreducible integrable representation of twisted toroidal EALAs with finite dimensional weight spaces where center acts non-trivially. The proof of our classification problem runs parallel to [2] where they consider full twisted TLA. Our twisted toroidal EALA is a proper subalgebra of full twisted EALA. Since we working with smaller algebra some of the results used in [2] won't work here. So we use differnt techniques; notably results from [5] and [15] are very special for divergence zero vector fields.We now explain our plan of classification result in more detail. In Section 2 we start with basic definitions which leads up to the definition of twisted toroidal EALA T . In Section 3 we define root space decomposition and using it define a natural triangular decomposition T = T − ⊕ T 0 ⊕ T + . We fix an irreducible integrable module V of T with finite dimensional weight spaces where K 0 acts non-trivially and K i act trivially for 1 ≤ i ≤ n. This assumption is due to the fact that group GL(n + 1, Z) acts as automorphism on FTLA and leaves τ div invariant. Then using earlier work of [2] it follows that the space M = {v ∈ V |T + v = 0} is a non-zero T 0 irreducible module.The rest of the paper revolves around M as we try to decode the structure of

Classification of irreducible integrable modules for extended affine Lie algebras with center acting trivially

“…Choose the i-th piece of g ′ ss and choose the proection of the map π, say π i onto it. Doing the same calculation as in [2] we will get π i (S ′ n (m)) = 0 and a r I i = 1 for all r ∈ Γ. Now suppose there is atleast two pieces, say i-th and j-th piece is there.…”

“…We know by Theorem 6 that the representation factors through l n copies, for some positive integer l. we will prove here that l = 1. The proof is same as [2]. Choose the i-th piece of g ′ ss and choose the proection of the map π, say π i onto it.…”

“…In this section we will prove that Ω An(m) /d An(m) acts trivially on V . The following two lemmas are proved in [2] as well as in [7] Proposition 4.1. There are only finitely many h( 0 Proof.…”

“…K n } acts trivially on V , then the whole central extension part Z = Ω An(m) /d An(m) acts trivially on V . In Section 5, to classify the modules we prove an important result Proposition 5.1 (2). We have used a new method to prove this.…”

“…This Lie algebra is called twisted toroidal extended affine Lie algebra, we denote it by τ . In [2], S.E. Rao, S. Sharma, P. Batra classified irreducible integrable modules for τ with finite dimensional weight spaces, where the zero degree central operators act nontrivially.…”