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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

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).


[1] F.A. Berezin, "Introduction to superanalysis" , Reidel (1987) (Translated from Russian)
[2] D.A. Leites (ed.) , Seminar on super-manifolds , Kluwer (1990)



[a1] F.A. Berezin, M.A. Shubin, "The Schrödinger equation" , Kluwer (1991) (Translated from Russian) (Supplement 3: D.A. Leites, Quantization and supermanifolds)
How to Cite This Entry:
Super-space. D.A. Leites (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098