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.
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
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 R0 (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 R0. 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
- The Berezinian of X is always a unit in the ring R0.
- where denotes the supertranspose of X.
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.
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- 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, 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
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