The theory of Vogan diagrams, which are Dynkin diagrams with an overlay of certain additional information, allows one to give a rapid classification of finitedimensional real semisimple Lie algebras and to make use of this classification in practice. This paper develops a corresponding theory of Vogan diagrams for "almost compact" real forms of indecomposable nontwisted affine Kac-Moody Lie algebras. In this case also, the equivalence classes of Vogan diagrams correspond to the isomorphism classes of almost compact real forms. Although the real forms of such algebras had already been classified, the theory of Vogan diagrams introduces invariants for such algebras and makes it possible to locate a given real form within the classification.

a b s t r a c tInspired by recent activities on Whittaker modules over various (Lie) algebras, we describe a general framework for the study of Lie algebra modules locally finite over a subalgebra. As a special case, we obtain a very general set-up for the study of Whittaker modules, which includes, in particular, Lie algebras with triangular decomposition and simple Lie algebras of Cartan type. We describe some basic properties of Whittaker modules, including a block decomposition of the category of Whittaker modules and certain properties of simple Whittaker modules under some rather mild assumptions. We establish a connection between our general set-up and the general set-up of Harish-Chandra subalgebras in the sense of Drozd, Futorny and Ovsienko. For Lie algebras with triangular decomposition, we construct a family of simple Whittaker modules (roughly depending on the choice of a pair of weights in the dual of the Cartan subalgebra), describe their annihilators, and formulate several classification conjectures. In particular, we construct some new simple Whittaker modules for the Virasoro algebra. Finally, we construct a series of simple Whittaker modules for the Lie algebra of derivations of the polynomial algebra, and consider several finite-dimensional examples, where we study the category of Whittaker modules over solvable Lie algebras and their relation to Koszul algebras.

A Vogan diagram is actually a Dynkin diagram with some additional structure. This paper develops theory of Vogan diagrams for "almost compact" real forms of indecomposable nontwisted affine Kac-Moody Lie algebras. Here, the equivalence classes of Vogan diagrams are in one-one correspondence with the isomorphism classes of almost compact real forms.  2002 Elsevier Science (USA)

This paper describes finite-dimensional irreducible representations of "twisted multi-loop Lie algebras." These representations are given in terms of the representations of finite-dimensional semisimple Lie algebras.  2004 Elsevier Inc. All rights reserved. IntroductionThe purpose of this paper is to classify irreducible finite-dimensional modules for twisted multi-loop Lie algebras. Let V be a vector space over the complex numbers andbe the Laurent polynomials ring in n variables t 1 , t 2 , . . . , t n . Let L(V ) = V ⊗ C L 1 and let v ⊗ t m = vt m 1 1 t m 2 2 . . . t m n n for m = (m 1 , m 2 , . . . , m n ) ∈ Z n and t m = t m 1 1 t m 2 2 . . . t m n n . Let g be a simple finite-dimensional Lie algebra over the complex numbers C. Then L(g) can be made into a Lie algebra by the bracket operation defined in Section 1. The universal central extension of L(g) is called a toroidal Lie algebra. A presentations of the toroidal Lie algebras are given in [6]. Representations of toroidal Lie algebras are studied in [1,5,6].Let µ be a diagram automorphism of g satisfying µ k = Id and let be a primitive kth root of unity. We extend µ to an automorphism of L(g) defined by

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