Some Studies on Infinite-Dimensional Lie(Super) Algebra

In this thesis, we study some results on infinite dimensional Lie algebras. Total thesis is divided into three parts, i.e., on first part we have determined untwisted affine Kac-Moody symmetric spaces, second part is devoted towards embedding of hypebolic Kac-Moody superalgberas and in the final part we study some branching laws for certain infinite dimensional reductive pair of Lie algebras. Symmetric spaces associated with Lie algebras and Lie groups which are Reimannian manifolds have recently got a lot of attention in various branches of physics and mathematics. Their infinite dimensional counterpart have recently been discovered which are affine Kac-Moody symmetric spaces. We have (algebraically) explicitly computed the affine Kac-Moody symmetric spaces associated with affine Kac-Moody algebras A (1) 1 ,A (1) 2 and A (2) 2 . We have also computed all the affine untwisted Kac-Moody symmetric spaces starting from the Vogan diagrams of the affine untwisted classical Kac-Moody Lie algebras. Root systems and Dynkin diagrams play a vital role in understanding and explaining the structure of corresponding algebras and superalgebras. Here through the help of the Dynkin diagrams and root systems we have given a super symmetric version of a theorem by S. Viswanath for hyperbolic Kac-Moody superalgebras. We have shown that HD(4,1) hyperbolic Kac-Moody superalgbera of rank 6 contains every simplylaced Kac-Moody subalgebra with degenerate odd root as a Lie subalgebra. Branching law is a classical problem in the representation theory of finite dimensional Lie algebras. Let g be a complex Lie algebra, g 0 be the Lie subalgebra of g andV be irreducible g-module then, V is no longer an irreducible g 0 -module. A branching law amounts to a decomposition of V into irreducible g 0 -module. However such a decomposition does not exist necessarily. The branching laws are understandable to some extent, in some nice setting (when g and g 0 are semisimple and V is finite dimensional). But for classical pairs (g,g 0 ) such as (gln ,gln−1 ), (son, son−1) etc. branching laws are explicitly known. Since each classical Lie algebra g fits into a descending family of classical algebras, the irreducible representations of g can be studied inductively. Here we have studied some branching laws for certain pairs (g,g 0 ) of infinite dimensional Lie algebras which are inductive limit of finite dimensional reductive Lie algebras