A Study on Kac-Moody Superalgebras

Abstract

Hyperbolic Kac-Moody algebras could account for a variety of problems in the realms of string theory (E_10), duality properties of super symmetric gauge theory and two dimensional field theories.The all possible hyperbolic Dynkin diagrams between rank 3 and 10 are already classified (finite in no.) along with rank 2 hyperbolic algebra (infinite in no.).It is natural to visualize, evaluate and interpret the possible consequences of super symmetric extension of these algebras. The constructs so generated are the hyperbolic Kac-Moody superalgebras.Recently these algebras have been classified by many authors including us where it is shown that they are limited in rank and now the maximum rank being 6 and they are finite in number(for rank>2). In physical application point of view not only hyperbolic Kac-Moody algebras (super) are important but their subalgebras also form an interesting part of such type of studies. These types of studies led to one important fact that the rank 10(highest rank) hyperbolic Kac-Moody algebraE_10containsevery simply laced hyperbolic Kac-Moody algebras as Liesub algebra. Extending such studies to hyperbolic Kac-Moody superalgebras, in this paper we have shown that every simply laced hyperbolic Kac-Moody algebra are embedded in three no. of rank 6 (highest for super case) simply laced hyperbolic kac-Moody superalgebras. It has been corroborated by proving some theorems and lemmas in supersymmetric case. However we have restricted our studies in distinguished basis of such algebras only. One can go from distinguish basis to other basis very easily by extended weyl group.