Algebraic Aspects of the Berezinian

Abstract

Many concepts of linear algebra can be generalized to the Z/2-graded setting, leading to linear superalgebra. Often, a formulation in terms of category theory facilitates this passage, and this e.g. provides an invariant description of the supertrace of an endomorphism T : V → V of a super vector space. However, it is not so straightforward to describe the superdeterminant, also known as Berezinian, in abasis-independent way. In this thesis we look at a characterization of the Berezinian, given by Deligneand Morgan, in terms of homological algebra. It generalizes the description of theordinary determinant via the induced action on the top exterior power of a vectorspace. After introducing super linear algebra, we explain the invariant description,and illustrate it by explicitly working it out for some examples.