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)