A class of integrable modules for the core of EALA coordinatized by quantum tori

Abstract

Extended affine Lie algebras (EALA) are higher dimensional generalizations of affine Kac–Moody Lie algebras introduced in [19]. They have been further studied in [1,2,5]. Toroidal Lie algebras which are universal central extensions of G ⊗ C[t±1 1 , . . . , t±1 n ] (G simple finite dimensional Lie algebra) and are prime examples of cores of EALAs which are studied in [4,7,14–16,22,24,25]. There are many EALAs which allow not only Laurent polynomial algebra C[t±1 1 , . . . , t±1 n ] as co-ordinate algebra but also quantum tori, Jordan tori and the octonians tori as co-ordinated algebras depending on type of the Lie algebra (see [1,3,5,6,26]). The structure theory of the EALA of type Ad−1 is tied up with Lie algebra gld(C)⊗Cq where Cq is the quantum torus (see Section 1). Quantum tori defined in [21] are non-commutative analogue of Laurent polynomial algebras. In this paper we study integrable modules for the core of the EALA corresponding to gld(C)⊗ Cq which we call τ . We will also add derivations d1, d2 to τ and call it τ . Here Cq is two variable quantum torus defined as the ring of non-commutating Laurent polynomials C[t±1 1 , t±1 2 ] with relation to t1t2 = qt2t1. Note that if q is generic then τ is an EALA. We classify (Theorem 5.2) irreducible integrable modules (see Definition 1.9) for τ where the zero degree center acts non-trivially. We prove that any such module is isomorphic to a highest weight module or a lowest module up to an automorphism (see Section 5). Let ∆ and ∆aff be finite and affine root system of sld(C) and its affinization. Consider ∆+ = ∆aff ⊕ Zδ2 and consider the subalgebra B corresponding to ∆+ (see Section 1). By highest weight modules we mean a module with a weight vector killed by B and the highest weight ψ :H →C[t, t−1] given as in Theorem 4.1. For any such ψ we define a highest weight module.