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| In [[mathematics]] and [[theoretical physics]], the '''Berezinian''' or '''superdeterminant''' is a generalization of the [[determinant]] to the case of [[supermatrix|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]].
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| ==Definition==
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| The Berezinian is uniquely determined by two defining properties:
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| *<math>\operatorname{Ber}(XY) = \operatorname{Ber}(X)\operatorname{Ber}(Y)</math>
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| *<math>\operatorname{Ber}(e^X) = e^{\operatorname{str(X)}}\,</math>
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| where str(''X'') denotes the [[supertrace]] of ''X''. Unlike the classical determinant, the Berezinian is defined only for invertible supermatrices.
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| The simplest case to consider is the Berezinian of a supermatrix with entries in a [[field (mathematics)|field]] ''K''. Such supermatrices represent [[linear transformation]]s of a [[super vector space]] over ''K''. A particular even supermatrix is a [[block matrix]] of the form
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| :<math>X = \begin{bmatrix}A & 0 \\ 0 & D\end{bmatrix}</math> | |
| 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
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| :<math>\operatorname{Ber}(X) = \det(A)\det(D)^{-1}</math>
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| For a motivation of the negative exponent see the [[Berezin integration#Multiple variables|substitution formula]] in the odd case.
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| More generally, consider matrices with entries in a [[supercommutative algebra]] ''R''. An even supermatrix is then of the form
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| :<math>X = \begin{bmatrix}A & B \\ C & D\end{bmatrix}</math>
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| 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''<sub>0</sub> (the [[even subalgebra]] of ''R''). In this case the Berezinian is given by
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| :<math>\operatorname{Ber}(X) = \det(A-BD^{-1}C)\det(D)^{-1}</math>
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| or, equivalently, by
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| :<math>\operatorname{Ber}(X) = \det(A)\det(D-CA^{-1}B)^{-1}.</math>
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| These formulas are well-defined since we are only taking determinants of matrices whose entries are in the commutative ring ''R''<sub>0</sub>. The matrix
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| : <math> D-CA^{-1}B \, </math>
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| is known as the [[Schur complement]] of ''A'' relative to <math>\begin{bmatrix} A & B \\ C & D \end{bmatrix}.</math>
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| 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
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| :<math>J = \begin{bmatrix}0 & I \\ -I & 0\end{bmatrix}.</math>
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| Then the Berezinian of ''X'' is defined as
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| :<math>\operatorname{Ber}(X) = \operatorname{Ber}(JX) = \det(C-DB^{-1}A)\det(-B)^{-1}.</math>
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| ==Properties==
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| *The Berezinian of ''X'' is always a [[unit (ring theory)|unit]] in the ring ''R''<sub>0</sub>.
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| *<math>\operatorname{Ber}(X)^{-1} = \operatorname{Ber}(X^{-1})</math>
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| *<math>\operatorname{Ber}(X^T) = \operatorname{Ber}(X)</math> where <math>X^T</math> denotes the supertranspose of ''X''.
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| *<math>\operatorname{Ber}(X\oplus Y) = \operatorname{Ber}(X)\mathrm{Ber}(Y)</math>
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| ==Berezinian module==
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| 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.
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| 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 functor|ext]] module
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| :<math>Ext_{A}^p (R,A)</math>
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| (which has dimension (1,0) if ''q'' is even and dimension (0,1) if ''q'' is odd))
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| as multiplication by the Berezianian.
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| ==See also==
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| *[[Berezin integral|Berezin integration]]
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| ==References==
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| *{{Citation | last1=Berezin | first1=Feliks Aleksandrovich | title=The method of second quantization | origyear=1965 | url=http://books.google.com/books?id=fAlRAAAAMAAJ | publisher=[[Academic Press]] | location=Boston, MA | series= Pure and Applied Physics | isbn=978-0-12-089450-5 | mr=0208930 | year=1966 | volume=24}}
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| *{{Citation | last1=Deligne | first1=Pierre | author1-link=Pierre Deligne | last2=Morgan | first2=John W. | editor1-last=Deligne | editor1-first=Pierre | editor1-link=Pierre Deligne | editor2-last=Etingof | editor2-first=Pavel | editor3-last=Freed | editor3-first=Daniel S. | editor4-last=Jeffrey | editor4-first=Lisa C. | editor5-last=Kazhdan | editor5-first=David | editor6-last=Morgan | editor6-first=John W. | editor7-last=Morrison | editor7-first=David R. | editor8-last=Witten. | editor8-first=Edward | title=Quantum fields and strings: a course for mathematicians, Vol. 1 | url=http://books.google.com/books?id=TQIsyvw1KnsC&pg=PA41 | publisher=[[American Mathematical Society]] | location=Providence, R.I. | isbn=978-0-8218-1198-6 | mr=1701597 | year=1999 | chapter=Notes on supersymmetry (following Joseph Bernstein) | pages=41–97}}
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| *{{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}}
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| [[Category:Super linear algebra]]
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| [[Category:Determinants]]
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