# 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)