superdeterminant

The superdeterminant, also known as the Berezinian (often abbreviated Ber or sdet), is a generalization of the ordinary determinant to square supermatrices. It arises in the context of Z2-graded vector spaces and plays a key role in integration on superspaces and in the representation theory of linear supergroups.

A square supermatrix can be written in a block form M = [A B; C D], where A and D are even-even blocks and B and C are odd blocks. The Berezinian is defined for invertible M by formulas that parallel the ordinary determinant but involve Schur complements. If A is invertible, then Ber(M) = det(A) det(D − C A^{-1} B)^{-1}. If D is invertible, Ber(M) = det(D) det(A − B D^{-1} C)^{-1}. These expressions are equal when both A and D are invertible, and Ber is defined for all invertible supermatrices.

Key properties include Ber(MN) = Ber(M) Ber(N) (multiplicativity) and Ber(I) = 1. In the diagonal-block case B = C = 0, Ber(M) reduces to det(A) det(D)^{-1}, mirroring the product structure of determinants on block-diagonal matrices. The Berezinian is a nonzero rational function of the matrix entries, and it can be viewed as the appropriate Jacobian factor for change of variables in integrals over superspaces.

Applications of the superdeterminant appear in supergeometry, quantum field theory with fermionic variables, and the theory of integration on supermanifolds. It is named after Felix Berezin, who introduced the concept in the 1960s.