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Article

Cohomological Approach to the Graded Berezinian

Journal of Noncommutative Geometry. 2015. Vol. 9. No. 2. P. 543-565.
Covolo T.

We develop the theory of linear algebra over a (Z2)n-commutative algebra (nN), 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.