Cohomological Approach to the Graded Berezinian

TL;DR

This paper extends linear algebra to (Z_2)^n-graded commutative algebras, introducing cohomological notions of trace and determinant that generalize classical and super cases, with applications to Clifford and quaternion algebras.

Contribution

It develops a cohomological framework for linear algebra over (Z_2)^n-graded algebras, generalizing the Berezinian to a broader class of graded structures.

Findings

Defined cohomological trace and determinant for graded algebras

Unified classical determinant and Berezinian within a single framework

Applied theory to Clifford and quaternion algebras

Abstract

We develop the theory of linear algebra over a (Z_2)^n-commutative algebra (n in N), which includes the well-known super linear algebra as a special case (n=1). Examples of such graded-commutative algebras are the Clifford algebras, in particular the quaternion algebra H. Following a cohomological approach, we introduce analogues of the notions of trace and determinant. Our construction reduces in the classical commutative case to the coordinate-free description of the determinant by means of the action of invertible matrices on the top exterior power, and in the supercommutative case it coincides with the well-known cohomological interpretation of the Berezinian.