Super-space
A vector space over a field
endowed with a
-grading
. The elements of the spaces
and
are said to be even and odd, respectively; for
, the parity
is defined to be
. Each super-space
has associated to it another super-space
such that
. The pair
, where
,
, is called the dimension of the super-space
. The field
is usually considered as a super-space of dimension
.
For two super-spaces and
, the structure of a super-space on the spaces
,
,
, etc., is defined naturally. In particular, a linear mapping
is even if
, and odd if
. A homogeneous bilinear form
is said to be symmetric if
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and skew-symmetric if
![]() |
All these concepts apply equally to -graded free modules
over an arbitrary commutative superalgebra
. The basis in
is usually selected so that its first vectors are even and its last ones odd. Any endomorphism
of the module
is denoted in this basis by a block matrix
![]() |
where ,
, such that if
is even, then
and
consist of even elements and
and
consist of odd elements, whereas if
is odd, then
and
consist of odd elements and
and
consist of even elements (in the former case the matrix
is even, in the latter, odd).
References
[1] | F.A. Berezin, "Introduction to superanalysis" , Reidel (1987) (Translated from Russian) |
[2] | D.A. Leites (ed.) , Seminar on super-manifolds , Kluwer (1990) |
Comments
References
[a1] | F.A. Berezin, M.A. Shubin, "The Schrödinger equation" , Kluwer (1991) (Translated from Russian) (Supplement 3: D.A. Leites, Quantization and supermanifolds) |
Super-space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Super-space&oldid=49616