Cohomological approach to the graded Berezinian

Abstract

We develop the theory of linear algebra over a $({\mathbb{Z}}_2{)}^n$-commutative algebra ($n\in \mathbb{N}$), 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 $\mathbb{H}$. 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.