1207.2962 (Tiffany Covolo)

Cohomological Approach to the Graded Berezinian    [PDF]

Tiffany Covolo

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, whereas in the supercommutative case it coincides with the cohomological description of the Berezinian due to Manin.

View original: http://arxiv.org/abs/1207.2962