Berezinian

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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.

[edit] 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.

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}.

[edit] Properties

  • The Berezinian of X is always a unit in the ring R0.
  • \operatorname{Ber}(X)^{-1} = \operatorname{Ber}(X^{-1})
  • \operatorname{Ber}(X^T) = \operatorname{Ber}(X) where XT denotes the supertranspose of X.
  • \operatorname{Ber}(X\oplus Y) = \operatorname{Ber}(X)\mathrm{Ber}(Y)

[edit] See also