On representations and structures of infinite dimensional Lie algebras

Abstract

newline In this thesis, we study two aspects of infinite dimensional Lie algebras. newlineIn the first part, we study the fusion product modules for current Lie algebras of type A 2 . newlineFusion products of finite-dimensional cyclic modules, that were defined in [23], form an newlineimportant class of graded representations of current Lie algebras. In [16], a family of finite- newlinedimensional indecomposable graded representations of the current Lie algebra called the newlineChari-Venkatesh(CV) modules, were introduced via generators and relations, and it was newlineshown that these modules are related to fusion products. We study a class of CV modules for newlinecurrent Lie algebras of type A 2 . By constructing a series of short exact sequences, we obtain newlinea graded decomposition for them and show that they are isomorphic to fusion products of newlinetwo finite-dimensional irreducible modules for current Lie algebras of sl 3 . Further, using newlinethe graded character of these CV-modules, we obtain an algebraic characterization of the newlineLittlewood-Richardson coefficients that appear in the decomposition of tensor products of newlineirreducible sl 3 (C)-modules. newlineIn the second part, we study the free root spaces of Borcherds-Kac-Moody Lie superalgebras. newlineLet L be a Borcherds-Kac-Moody Lie superalgebra (BKM superalgebra in short) with the newlineassociated graph G. Any such L is constructed from a free Lie superalgebra by introducing newlinethree different sets of relations on the generators: (1) Chevalley relations, (2) Serre relations, newlineand (3) Commutation relations coming from the graph G. By Chevalley relations we get a newlinetriangular decomposition L = n + and#8853; h and#8853; n and#8722; and each roots space L and#945; is either contained in newlinen + or n and#8722; . In particular, each L and#945; involves only the relations (2) and (3). We study the rootxii newlinespaces of L which are independent of the Serre relations. We call these roots the free roots of newlineL. Since these root spaces involve only commutation relations coming from the graph, G newlinewe can study them combinatorially.We construct two different bases for these root spaces newlineof, L using