Quasi Hopf superalgebras and their dual structures

Abstract

This dissertation investigates the Z2-graded generalizations of Drinfeld's quasi Hopf algebras and their dual structures. Definitions are presented in terms of commutative diagrams which simplifies the process of dualizing the structure.We look in detail at the concept of twisting and construct invariants and central elements for these Z2-graded quasi Hopf structures. Following this, we present an explicit construction of the Drinfeld twist which transforms the structure of a quasi Hopf (super)algebra into another which can be obtained by straightforward application of the antipode. In general these two structures will not coincide, except in the case of ordinary Hopf (super)algebras.After presenting the basic theory of co-quasi Hopf superalgebras, we look at two consequences of the definition, one being the quasi function algebras, which are non- associative generalizations of (quantized) function algebras, the other consequence being a generalization of Lie algebras. Given this second outcome, we then look at an alternative definition of co-twisting on these new structures, which turn out to be invariant under this general type of co-twist. We then look at some examples. This provides a new perspective on the problem of generalizing the theory of Lie algebras