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

The Berezinian is uniquely determined by two defining properties:

*$\operatorname(XY) = \operatorname(X)\operatorname(Y)$
*$\operatorname(e^X) = e^\,$

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
:$X = \beginA & 0 \\ 0 & D\end$
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
:$\operatorname(X) = \det(A)\det(D)^$

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
:$X = \beginA & B \\ C & D\end$
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

:$\operatorname(X) = \det(A-BD^C)\det(D)^$

or, equivalently, by

:$\operatorname(X) = \det(A)\det(D-CA^B)^.$

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

: $D-CA^B \,$

is known as the Schur complement of ''A'' relative to $\begin A & B \\ C & D \end.$

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
:$J = \begin0 & I \\ -I & 0\end.$
Then the Berezinian of ''X'' is defined as
:$\operatorname(X) = \operatorname(JX) = \det(C-DB^A)\det(-B)^.$

Properties

*The Berezinian of $X$ is always a unit in the ring ''R''₀.
*$\operatorname(X^) = \operatorname(X)^$
*$\operatorname(X^) = \operatorname(X)$ where $X^$ denotes the supertranspose of $X$.
*$\operatorname(X\oplus Y) = \operatorname(X)\mathrm(Y)$

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
:$Ext_^p (R,A)$
(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

References

*
*
*{{Citation , last1=Manin , first1=Yuri Ivanovich , author1-link=Yuri Ivanovich Manin , title=Gauge Field Theory and Complex Geometry , publisher=Springer-Verlag , location=Berlin, New York , edition=2nd , isbn=978-3-540-61378-7 , year=1997

Super linear algebra
Determinants