# Super-group

Lie super-group

A group object in the category of super-manifolds. A super-group ${\mathcal G}$ is defined by a functor ${\mathcal G}$ from the category of commutative superalgebras into the category of groups. Lie's theorems (cf. Lie theorem) are transferred to super-groups, a fact that gives the correspondence between super-groups and finite-dimensional Lie superalgebras.

### Examples.

1) The super-group $\mathop{\rm GL} _ {n\mid m}$ is defined by the functor $C \mapsto \mathop{\rm GL} _ {n\mid m} ( C )$ into groups of even invertible matrices from $M _ {n\mid m} ( C)$( see Super-space), i.e. of matrices in the form

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

where $X, T$ are invertible matrices of orders $n, m$ over $C _ {\overline{0}\; }$, while $Y, Z$ are matrices over $C _ {\overline{1}\; }$. A homomorphism $\mathop{\rm GL} _ {n\mid m} ( C) \rightarrow C _ {\overline{0}\; } ^ \star$ is defined by the formula

$$\mathop{\rm Ber} \left ( \begin{array}{cc} X & Y \\ Z & T \\ \end{array} \right ) = \mathop{\rm det} ( X- YT ^ {-1} Z) \mathop{\rm det} T ^ {-1}$$

(the Berezinian);

2) $\mathop{\rm SL} _ {n\mid m} = \mathop{\rm Ker} \mathop{\rm Ber}$;

3) $\mathop{\rm OSp} _ {n\mid 2m} \subset \mathop{\rm GL} _ {n\mid 2m}$ and $\Pi _ {n} \subset \mathop{\rm GL} _ {n\mid m}$; they leave invariant an even, or odd, non-degenerate symmetric bilinear form.

To every super-group ${\mathcal G}$ and super-subgroup ${\mathcal H}$ of it there is related a super-manifold ${\mathcal G} / {\mathcal H}$, represented by a functor $C \mapsto {\mathcal G} ( C) / {\mathcal H} ( C)$. This super-manifold is a homogeneous space of ${\mathcal G}$.

#### References

 [1] Yu.I. Manin, "Gauge fields and complex geometry" , Springer (1988) (Translated from Russian) [2] F.A. Berezin, "Introduction to superanalysis" , Reidel (1987) (Translated from Russian) [3] D.A. Leites (ed.) , Seminar on supermanifolds , Kluwer (1990)
How to Cite This Entry:
Super-group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Super-group&oldid=50892
This article was adapted from an original article by D.A. Leites (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article