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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.

Contents

Definition

The Berezinian is uniquely determined by two defining properties:

  • Ber ⁡ ( X Y ) = Ber ⁡ ( X ) Ber ⁡ ( Y )
  • Ber ⁡ ( e X ) = e s t r ( 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 = [ A 0 0 D ]

    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

    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 = [ A B C D ]

    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

    Ber ⁡ ( X ) = det ( A − B D − 1 C ) det ( D ) − 1

    or, equivalently, by

    Ber ⁡ ( X ) = det ( A ) det ( D − C A − 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 − C A − 1 B

    is known as the Schur complement of A relative to [ A B C D ] .

    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 = [ 0 I − I 0 ] .

    Then the Berezinian of X is defined as

    Ber ⁡ ( X ) = Ber ⁡ ( J X ) = det ( C − D B − 1 A ) det ( − B ) − 1 .

    Properties

  • The Berezinian of X is always a unit in the ring R0.
  • Ber ⁡ ( X − 1 ) = Ber ⁡ ( X ) − 1
  • Ber ⁡ ( X s t ) = Ber ⁡ ( X ) where X s t denotes the supertranspose of X .
  • Ber ⁡ ( X ⊕ Y ) = Ber ⁡ ( X ) B e r ( 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

    E x t A p ( R , A )

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

    References

    Berezinian Wikipedia
    (Text) CC BY-SA