Super vector space - Knowlpedia

Definition

Let $k$ be a field. A super vector space over $k$ is a $\mathbb Z/2$ -graded vector space

V=V_{\bar 0}\oplus V_{\bar 1}.

Elements of $V_{\bar 0}$ are even, elements of $V_{\bar 1}$ are odd, and a nonzero element in either summand is homogeneous. The parity of a homogeneous element $v$ is $|v|\in\mathbb Z/2$ , determined by $v\in V_{|v|}$ .

Maps and dimensions

A homogeneous linear map $f:V\to W$ has parity $\epsilon$ when

f(V_{\bar i})\subseteq W_{\bar i+\epsilon}.

Thus even maps preserve parity and odd maps reverse it. Unless stated otherwise, morphisms between super vector spaces are even maps; odd maps belong to the graded internal Hom rather than the ordinary morphism set.

If both summands are finite-dimensional, the graded dimension is

\dim_{\mathrm{gr}}V =\dim V_{\bar0}\mid\dim V_{\bar1}.

The ordered pair, not its difference, determines the dimensions of the two summands. The categorical superdimension, namely the trace of the identity computed with the Koszul sign rule, is

\operatorname{sdim}(V) =\dim V_{\bar0}-\dim V_{\bar1}.

Some authors call either invariant “superdimension,” so specifying “graded” or “categorical” avoids an ambiguity.

Examples

Every ordinary vector space becomes a purely even super vector space by putting $V_{\bar1}=0$ . The exterior algebra $\Lambda U$ is a super vector space when exterior degree is reduced modulo $2$ :

(\Lambda U)_{\bar0}=\bigoplus_j\Lambda^{2j}U, \qquad (\Lambda U)_{\bar1}=\bigoplus_j\Lambda^{2j+1}U.

Characteristic and terminology

The decomposition itself makes sense in every characteristic. Throughout the super sign convention used in these knowls, $k$ has characteristic different from $2$ . In characteristic $2$ , the sign $(-1)^{|v||w|}$ cannot distinguish commuting from anticommuting odd elements, and several standard definitions need extra structure.

“Graded vector space” can refer to a grading by $\mathbb Z$ or another group. The adjective “super” specifically means a $\mathbb Z/2$ -grading.

References

V. S. Varadarajan, Supersymmetry for Mathematicians: An Introduction, American Mathematical Society, 2004. DOI record. Relevant: Chapter 1.
C. Carmeli, L. Caston, and R. Fioresi, Mathematical Foundations of Supersymmetry, European Mathematical Society, 2011. DOI record. Relevant: Chapter 1.