Supermatrix

In mathematics and theoretical physics, a supermatrix is a Z₂-graded analog of an ordinary matrix. Specifically, a supermatrix is a 2×2 block matrix with entries in a superalgebra (or superring). The most important examples are those with entries in a commutative superalgebra (such as a Grassmann algebra) or an ordinary field (thought of as a purely even commutative superalgebra).

Supermatrices arise in the study of super linear algebra where they appear as the coordinate representations of a linear transformations between finite-dimensional super vector spaces or free supermodules. They have important applications in the field of supersymmetry.

Definitions and notation

Let "R" be a fixed superalgebra (assumed to be unital and associative). Often one requires "R" be supercommutative as well (for essentially the same reasons as in the ungraded case).

Let "p", "q", "r", and "s" be nonnegative integers. A supermatrix of dimension ("r"|"s")×("p"|"q") is a matrix with entries in "R" that is partitioned into a 2×2 block structure:with "r"+"s" total rows and "p"+"q" total columns (so that the submatrix "X"₀₀ has dimensions "r"×"p" and "X"₁₁ has dimensions "s"×"q"). An ordinary (ungraded) matrix can be thought of as a supermatrix for which "q" and "s" are both zero.

A "square" supermatrix is one for which ("r"|"s") = ("p"|"q"). This means that not only is the unpartitioned matrix "X" square, but the diagonal blocks "X"₀₀ and "X"₁₁ are as well.

An even supermatrix is one for which diagonal blocks ("X"₀₀ and "X"₁₁) consist solely of even elements of "R" (i.e. homogeneous elements of parity 0) and the off-diagonal blocks ("X"₀₁ and "X"₁₀) consist solely of odd elements of "R".:An odd supermatrix is one for the reverse holds: the diagonal blocks are odd and the off-diagonal blocks are even.:If the scalars "R" are purely even there are no nonzero odd elements, so the even supermatices are the block diagonal ones and the odd supermatrices are the off-diagonal ones.

A supermatrix is homogeneous if it is either even or odd. The parity, |"X"|, of a nonzero homogeneous supermatrix "X" is 0 or 1 according to whether it is even or odd. Every supermatrix can be written uniquely as the sum of an even supermatrix and an odd one.

Algebraic structure

Supermatrices of compatible dimensions can be added or multiplied just as for ordinary matrices. These operations are exactly the same as the ordinary ones with the restriction that they are defined only when the blocks have compatible dimensions. One can also multiply supermatrices by elements of "R" (on the left or right), however, this operation differs from the ungraded case due to the presence of odd elements in "R".

Let "M"_{"r"|"s"×"p"|"q"}("R") denote the set of all supermatrices over "R" with dimension ("r"|"s")×("p"|"q"). This set forms a supermodule over "R" under supermatrix addition and scalar multiplication. In particular, if "R" is a superalgebra over a field "K" then "M"_{"r"|"s"×"p"|"q"}("R") forms a super vector space over "K".

Let "M"_"p"|"q"("R") denote the set of all square supermatices over "R" with dimension ("p"|"q")×("p"|"q"). This set forms a superring under supermatrix addition and multiplication. Furthermore, if "R" is a commutative superalgebra, then supermatrix multiplication is a bilinear operation, so that "M"_"p"|"q"("R") forms a superalgebra over "R".

Addition

Two supermatrices of dimension ("r"|"s")×("p"|"q") can be added just as in the ungraded case to obtain a supermatrix of the same dimension. The addition can be performed blockwise since the blocks have compatible sizes. It is easy to see that the sum of two even supermatrices is even and the sum of two odd supermatrices is odd.

Multiplication

One can multiply a supermatrix with dimensions ("r"|"s")×("p"|"q") by a supermatrix with dimensions ("p"|"q")×("k"|"l") as in the ungraded case to obtain a matrix of dimension ("r"|"s")×("k"|"l"). The multiplication can be performed at the block level in the obvious manner::Note that the blocks of the product supermatrix "Z" = "XY" are given by:$Z\_\{ij\}\; =\; X\_\{i0\}Y\_\{0j\}\; +\; X\_\{i1\}Y\_\{1j\}.,$If "X" and "Y" are homogeneous with parities |"X"| and |"Y"| then "XY" is homogeneous with parity |"X"| + |"Y"|. That is, the product of two even or two odd supermatrices is even while the product of an even and odd supermatrix is odd.

calar multiplication

Scalar multiplication for supermatrices is different than the ungraded case due to the presence of odd elements in "R". Let "X" be a supermatrix. Left scalar multiplication by α ∈ "A" is defined by:where the internal scalar multiplications are the ordinary ungraded ones and $hatalpha$ denotes the grade involution in "A". This is given on homogeneous elements by:$hatalpha\; =\; (-1)^C^t\; \backslash \; -(-1)^mathrm\{tr\}(X\_\{11\}),$where tr denotes the ordinary trace.

If "R" is supercommutative, the supertrace satisfies the identity:$mathrm\{str\}(XY)\; =\; (-1)^mathrm\{str\}(YX),$for homogeneous supermatrices "X" and "Y".

Berezinian

The Berezinian (or superdeterminant) of a square supermatrix is the Z₂-graded analog of the determinant. The Berezinian is only well-defined on even, invertible supermatrices over a commutative superalgebra "R". In this case it is given by the formula:$mathrm\{Ber\}(X)\; =\; det(X\_\{00\}\; -\; X\_\{01\}X\_\{11\}^\{-1\}X\_\{10\})det(X\_\{11\})^\{-1\}.$where det denotes the ordinary determinant (of square matrices with entries in the commutative algebra "R"₀).

The Berezinian satisfies similar properties to the ordinary determinate. In particular, it is multiplicative and invariant under the supertranspose. It is related to the supertrace by the formula:$mathrm\{Ber\}(e^X)\; =\; e^\{mathrm\{str(X).,$

References

*cite book | first = V. S. | last = Varadarajan | year = 2004 | title = Supersymmetry for Mathematicians: An Introduction | series = Courant Lecture Notes in Mathematics 11 | publisher = American Mathematical Society | id = ISBN 0-8218-3574-2
*cite conference | first = Pierre | last = Deligne | coauthors = John W. Morgan | title = Notes on Supersymmetry (following Joseph Bernstein) | booktitle = Quantum Fields and Strings: A Course for Mathematicians | volume = 1 | pages = 41–97 | publisher = American Mathematical Society | year = 1999 | id = ISBN 0-8218-2012-5