# Berezinian

In mathematics and theoretical physics, the **Berezinian** or **superdeterminant** is a generalization of the determinant to the case of supermatrices. The name is for Felix Berezin; the Berezinian plays a role analogous to the determinant when considering coordinate changes for integration on a supermanifold.

## Definition[edit]

The Berezinian is uniquely determined by two defining properties:

where str(*X*) denotes the supertrace of *X*. Unlike the classical determinant, the Berezinian is defined only for invertible supermatrices.

The simplest case to consider is the Berezinian of a supermatrix with entries in a field *K*; such supermatrices represent linear transformations of a super vector space over *K*. A particular even supermatrix is a block matrix of the form

Such a matrix is invertible if and only if both *A* and *D* are invertible matrices over *K*; the Berezinian of *X* is given by

For a motivation of the negative exponent see the substitution formula in the odd case.

More generally, consider matrices with entries in a supercommutative algebra *R*. An even supermatrix is then of the form

where *A* and *D* have even entries and *B* and *C* have odd entries. Such a matrix is invertible if and only if both *A* and *D* are invertible in the commutative ring *R*_{0} (the even subalgebra of *R*). In this case the Berezinian is given by

or, equivalently, by

These formulas are well-defined since we are only taking determinants of matrices whose entries are in the commutative ring *R*_{0}. The matrix

is known as the Schur complement of *A* relative to

An odd matrix *X* can only be invertible if the number of even dimensions equals the number of odd dimensions. In this case, invertibility of *X* is equivalent to the invertibility of *JX*, where

Then the Berezinian of *X* is defined as

## Properties[edit]

- The Berezinian of is always a unit in the ring
*R*_{0}. - where denotes the supertranspose of .

## Berezinian module[edit]

The determinant of an endomorphism of a free module *M* can be defined as the induced action on the 1-dimensional highest exterior power of *M*. In the supersymmetric case there is no highest exterior power, but there is a still a similar definition of the Berezinian as follows.

Suppose that *M* is a free module of dimension (*p*,*q*) over *R*. Let *A* be the (super)symmetric algebra *S**(*M**) of the dual *M** of *M*. Then an automorphism of *M* acts on the ext module

(which has dimension (1,0) if *q* is even and dimension (0,1) if *q* is odd))
as multiplication by the Berezianian.

## See also[edit]

## References[edit]

- Berezin, Feliks Aleksandrovich (1966) [1965],
*The method of second quantization*, Pure and Applied Physics,**24**, Boston, MA: Academic Press, ISBN 978-0-12-089450-5, MR 0208930 - Deligne, Pierre; Morgan, John W. (1999), "Notes on supersymmetry (following Joseph Bernstein)", in Deligne, Pierre; Etingof, Pavel; Freed, Daniel S.; Jeffrey, Lisa C.; Kazhdan, David; Morgan, John W.; Morrison, David R.; Witten., Edward (eds.),
*Quantum fields and strings: a course for mathematicians, Vol. 1*, Providence, R.I.: American Mathematical Society, pp. 41–97, ISBN 978-0-8218-1198-6, MR 1701597 - Manin, Yuri Ivanovich (1997),
*Gauge Field Theory and Complex Geometry*(2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-61378-7