Superinvolutions of associative superalgebras

Abstract

In the first part of this thesis, we study the existence of superinvolutions of both kinds on finite dimensional central simple superalgebras. We study the group of automorphisms of a central simple associative superalgebra with superinvolution, and then study the group of automorphisms for two examples in detail. In the second part, we give the definition of projective supermodules; it is used to study separable superalgebras over super-commutative super-rings. We prove that a central separable superalgebra over a given field is a finite dimensional central simple superalgebra over the field and vice versa. We also present the fundamental theory of separable superalgebras, with the attendant theory of the Brauer-Wall group of supercommutative super-ring. Finally, we give the definition of superinvolutions of central separable superalgebras, and then extend two results on involutions of central separable algebras proved by D. J. Saltman.