Berezinian Explained

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

The Berezinian is uniquely determined by two defining properties:

\operatorname{Ber}(XY) = \operatorname{Ber}(X)\operatorname{Ber}(Y) \operatorname{Ber}(e ^{X)} = e ^{\operatorname{str(X)} }\,

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

X = \begin{bmatrix}A & 0 \ 0 & D\end{bmatrix} 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\operatorname{Ber}(X) = \det(A)\det(D) ^{-1}

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

X = \begin{bmatrix}A & B \ C & D\end{bmatrix} 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\operatorname{Ber}(X) = \det(A - BD ^{-1} C)\det(D) ^{-1}

or, equivalently, by

\operatorname{Ber}(X) = \det(A)\det(D - CA ^{-1} B) ^{-1} .

These formulas are well-defined since we are only taking determinants of matrices whose entries are in the commutative ring R₀. The matrix

D - CA ^{-1} B

is known as the Schur complement of A relative to

\begin{bmatrix} A & B \ C & D \end{bmatrix}.

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

J = \begin{bmatrix} 0 & I \ - I & 0 \end{bmatrix}. Then the Berezinian of X is defined as\operatorname{Ber}(X) = \operatorname{Ber}(JX) = \det(C - DB ^{-1} A)\det( - B) ^{-1} .

Properties

• The Berezinian of

X is always a unit in the ring R₀.\operatorname{Ber}(X ^{-1} ) = \operatorname{Ber}(X) ^{-1} \operatorname{Ber}(X ^{st} ) = \operatorname{Ber}(X) where X ^{st} denotes the supertranspose of X .\operatorname{Ber}(X ⊕ Y) = \operatorname{Ber}(X) Ber (Y)

Berezinian module

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

Ext ^{p} _{A} (R,A) (which has dimension (1,0) if q is even and dimension (0,1) if q is odd))as multiplication by the Berezinian.

See also

• Berezin integration