Super-space

A vector space $ V $ over a field $ k $ endowed with a $ \mathbf Z / 2 $-grading $ V = V _ {\overline{0} } \oplus V _ {\overline{1} } $. The elements of the spaces $ V _ {\overline{0} } $ and $ V _ {\overline{1} } $ are said to be even and odd, respectively; for $ x \in V _ {i} $, the parity $ p( x) $ is defined to be $ i $ $ ( i \in \mathbf Z / 2 = \{ \overline{0} , \overline{1} \} ) $. Each super-space $ V $ has associated to it another super-space $ \Pi ( V) $ such that $ \Pi ( V) _ {i} = V _ {i+ \overline{1} } $ $ ( i \in \mathbf Z / 2 ) $. The pair $ ( m, n) $, where $ m = \mathop{\rm dim} V _ {\overline{0} } $, $ n = \mathop{\rm dim} V _ {\overline{1} } $, is called the dimension of the super-space $ V $. The field $ k $ is usually considered as a super-space of dimension $ ( 1, 0) $.

For two super-spaces $ V $ and $ W $, the structure of a super-space on the spaces $ V \oplus W $, $ \mathop{\rm Hom} _ {k} ( V, W) $, $ V ^ \star $, etc., is defined naturally. In particular, a linear mapping $ \phi : V \rightarrow W $ is even if $ \phi ( V _ {i} ) \subset W _ {i} $, and odd if $ \phi ( V _ {i} ) \subset W _ {i+ \overline{1} } $. A homogeneous bilinear form $ \beta : V \otimes V \mapsto k $ is said to be symmetric if

$$ \beta ( y, x) = (- 1) ^ {p( x) p( y)+ p( \beta )( p( x)+ p( y)) } \beta ( x, y), $$

and skew-symmetric if

$$ \beta ( y, x) = -(- 1) ^ {p( x) p( y)+ p( \beta )( p( x)+ p( y)) } \beta ( x, y). $$

All these concepts apply equally to $ \mathbf Z / 2 $-graded free modules $ V $ over an arbitrary commutative superalgebra $ C $. The basis in $ V $ is usually selected so that its first vectors are even and its last ones odd. Any endomorphism $ \phi $ of the module $ V $ is denoted in this basis by a block matrix

$$ \alpha = \left ( \begin{array}{cc} X & Y \\ Z & T \\ \end{array} \right ) , $$

where $ X \in M _ {n} ( C) $, $ T \in M _ {m} ( C) $, such that if $ \phi $ is even, then $ X $ and $ T $ consist of even elements and $ Y $ and $ Z $ consist of odd elements, whereas if $ \phi $ is odd, then $ X $ and $ T $ consist of odd elements and $ Y $ and $ Z $ consist of even elements (in the former case the matrix $ \alpha $ 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)

References

[a1]	F.A. Berezin, M.A. Shubin, "The Schrödinger equation" , Kluwer (1991) (Translated from Russian) (Supplement 3: D.A. Leites, Quantization and supermanifolds)