In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
and
theoretical physics, a supermatrix is a Z
2-graded analog of an ordinary
matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
. Specifically, a supermatrix is a 2×2
block matrix
In mathematics, a block matrix or a partitioned matrix is a matrix that is '' interpreted'' as having been broken into sections called blocks or submatrices. Intuitively, a matrix interpreted as a block matrix can be visualized as the original mat ...
with entries in a
superalgebra (or
superring). The most important examples are those with entries in a
commutative superalgebra
In mathematics, a supercommutative (associative) algebra is a superalgebra (i.e. a Z2-graded algebra) such that for any two homogeneous elements ''x'', ''y'' we have
:yx = (-1)^xy ,
where , ''x'', denotes the grade of the element and is 0 or 1 ( ...
(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
In a supersymmetric theory the equations for force and the equations for matter are identical. In theoretical and mathematical physics, any theory with this property has the principle of supersymmetry (SUSY). Dozens of supersymmetric theories e ...
.
Definitions and notation
Let ''R'' be a fixed
superalgebra (assumed to be
unital and
associative
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement f ...
). 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
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
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 the 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 which 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
In mathematics, a block matrix or a partitioned matrix is a matrix that is '' interpreted'' as having been broken into sections called blocks or submatrices. Intuitively, a matrix interpreted as a block matrix can be visualized as the original mat ...
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
In mathematics, a supercommutative (associative) algebra is a superalgebra (i.e. a Z2-graded algebra) such that for any two homogeneous elements ''x'', ''y'' we have
:yx = (-1)^xy ,
where , ''x'', denotes the grade of the element and is 0 or 1 ( ...
, 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
:
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.
Scalar 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 ∈ ''R'' is defined by
:
where the internal scalar multiplications are the ordinary ungraded ones and
denotes the grade involution in ''R''. This is given on homogeneous elements by
:
Right scalar multiplication by α is defined analogously:
:
If α is even then
and both of these operations are the same as the ungraded versions. If α and ''X'' are homogeneous then α·''X'' and ''X''·α are both homogeneous with parity , α, + , ''X'', . Furthermore, if ''R'' is supercommutative then one has
:
As linear transformations
Ordinary matrices can be thought of as the coordinate representations of
linear maps between
vector spaces (or
free module
In mathematics, a free module is a module that has a basis – that is, a generating set consisting of linearly independent elements. Every vector space is a free module, but, if the ring of the coefficients is not a division ring (not a field in t ...
s). Likewise, supermatrices can be thought of as the coordinate representations of linear maps between
super vector spaces (or
free supermodule In mathematics, a supermodule is a Z2-graded module over a superring or superalgebra. Supermodules arise in super linear algebra which is a mathematical framework for studying the concept supersymmetry in theoretical physics.
Supermodules over a co ...
s). There is an important difference in the graded case, however. A homomorphism from one super vector space to another is, by definition, one that preserves the grading (i.e. maps even elements to even elements and odd elements to odd elements). The coordinate representation of such a transformation is always an ''even'' supermatrix. Odd supermatrices correspond to linear transformations that reverse the grading. General supermatrices represent an arbitrary ungraded linear transformation. Such transformations are still important in the graded case, although less so than the graded (even) transformations.
A
supermodule ''M'' over a
superalgebra ''R'' is ''free'' if it has a free homogeneous basis. If such a basis consists of ''p'' even elements and ''q'' odd elements, then ''M'' is said to have rank ''p'', ''q''. If ''R'' is supercommutative, the rank is independent of the choice of basis, just as in the ungraded case.
Let ''R''
''p'', ''q'' be the space of column supervectors—supermatrices of dimension (''p'', ''q'')×(1, 0). This is naturally a right ''R''-supermodule, called the ''right coordinate space''. A supermatrix ''T'' of dimension (''r'', ''s'')×(''p'', ''q'') can then be thought of as a right ''R''-linear map
:
where the action of ''T'' on ''R''
''p'', ''q'' is just supermatrix multiplication (this action is not generally left ''R''-linear which is why we think of ''R''
''p'', ''q'' as a ''right'' supermodule).
Let ''M'' be free right ''R''-supermodule of rank ''p'', ''q'' and let ''N'' be a free right ''R''-supermodule of rank ''r'', ''s''. Let (''e''
''i'') be a free basis for ''M'' and let (''f''
''k'') be a free basis for ''N''. Such a choice of bases is equivalent to a choice of isomorphisms from ''M'' to ''R''
''p'', ''q'' and from ''N'' to ''R''
''r'', ''s''. Any (ungraded) linear map
:
can be written as a (''r'', ''s'')×(''p'', ''q'') supermatrix relative to the chosen bases. The components of the associated supermatrix are determined by the formula
:
The block decomposition of a supermatrix ''T'' corresponds to the decomposition of ''M'' and ''N'' into even and odd submodules:
:
Operations
Many operations on ordinary matrices can be generalized to supermatrices, although the generalizations are not always obvious or straightforward.
Supertranspose
The supertranspose of a supermatrix is the Z
2-graded analog of the
transpose. Let
:
be a homogeneous (''r'', ''s'')×(''p'', ''q'') supermatrix. The supertranspose of ''X'' is the (''p'', ''q'')×(''r'', ''s'') supermatrix
:
where ''A''
''t'' denotes the ordinary transpose of ''A''. This can be extended to arbitrary supermatrices by linearity. Unlike the ordinary transpose, the supertranspose is not generally an
involution, but rather has order 4. Applying the supertranspose twice to a supermatrix ''X'' gives
:
If ''R'' is supercommutative, the supertranspose satisfies the identity
:
Parity transpose
The parity transpose of a supermatrix is a new operation without an ungraded analog. Let
:
be a (''r'', ''s'')×(''p'', ''q'') supermatrix. The parity transpose of ''X'' is the (''s'', ''r'')×(''q'', ''p'') supermatrix
:
That is, the (''i'',''j'') block of the transposed matrix is the (1−''i'',1−''j'') block of the original matrix.
The parity transpose operation obeys the identities
*
*
*
*
as well as
*
*
where ''st'' denotes the supertranspose operation.
Supertrace
The
supertrace of a square supermatrix is the Z
2-graded analog of the
trace
Trace may refer to:
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* Trace (Son Volt album), ''Trace'' (Son Volt album), 1995
* Trace (Died Pretty album), ''Trace'' (Died Pretty album), 1993
* Trace (band), a Dutch progressive rock band
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. It is defined on homogeneous supermatrices by the formula
:
where tr denotes the ordinary trace.
If ''R'' is supercommutative, the supertrace satisfies the identity
:
for homogeneous supermatrices ''X'' and ''Y''.
Berezinian
The
Berezinian (or
superdeterminant 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 ...
) of a square supermatrix is the Z
2-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
:
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 determinant. In particular, it is multiplicative and invariant under the supertranspose. It is related to the supertrace by the formula
:
References
*
*{{cite conference , first1 = Pierre , last1 = Deligne , first2 = John W. , last2=Morgan , title = Notes on Supersymmetry (following Joseph Bernstein) , book-title = Quantum Fields and Strings: A Course for Mathematicians , volume = 1 , pages = 41–97 , publisher = American Mathematical Society , year = 1999 , isbn = 0-8218-2012-5
Matrices
Super linear algebra