On representations and structures of infinite-dimensional Lie algebras

Abstract

In this thesis, we study two aspects of infinite dimensional Lie algebras. In the first part, we study the fusion product modules for current Lie algebras of type A 2 . Fusion products of finite-dimensional cyclic modules, that were defined in [23], form an important class of graded representations of current Lie algebras. In [16], a family of finite- dimensional indecomposable graded representations of the current Lie algebra called the Chari-Venkatesh(CV) modules, were introduced via generators and relations, and it was shown that these modules are related to fusion products. We study a class of CV modules for current Lie algebras of type A 2 . By constructing a series of short exact sequences, we obtain a graded decomposition for them and show that they are isomorphic to fusion products of two finite-dimensional irreducible modules for current Lie algebras of sl 3 . Further, using the graded character of these CV-modules, we obtain an algebraic characterization of the Littlewood-Richardson coefficients that appear in the decomposition of tensor products of irreducible sl 3 (C)-modules. In the second part, we study the free root spaces of Borcherds-Kac-Moody Lie superalgebras. Let L be a Borcherds-Kac-Moody Lie superalgebra (BKM superalgebra in short) with the associated graph G. Any such L is constructed from a free Lie superalgebra by introducing three different sets of relations on the generators: (1) Chevalley relations, (2) Serre relations, and (3) Commutation relations coming from the graph G. By Chevalley relations we get a triangular decomposition L = n + ⊕ h ⊕ n − and each roots space L α is either contained in n + or n − . In particular, each L α involves only the relations (2) and (3). We study the rootxii spaces of L which are independent of the Serre relations. We call these roots the free roots of L. Since these root spaces involve only commutation relations coming from the graph, G we can study them combinatorially.We construct two different bases for these root spaces of, L using combinatorics of Lyndon heaps and super Lyndon words. Finally, we relate the k-chromatic polynomial with root multiplicities of BKM superalgebras.