Super-group
Lie super-group
A group object in the category of super-manifolds (cf. Super-manifold). A super-group is defined by a functor
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 (cf. Superalgebra).
Examples.
1) The super-group is defined by the functor
into groups of even invertible matrices from
(see Super-space), i.e. of matrices in the form
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where are invertible matrices of orders
over
, while
are matrices over
. A homomorphism
is defined by the formula
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(the Berezinian);
2) ;
3) and
; they leave invariant an even, or odd, non-degenerate symmetric bilinear form.
To every super-group and super-subgroup
of it there is related a super-manifold
, represented by a functor
. This super-manifold is a homogeneous space of
.
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) |
Super-group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Super-group&oldid=49615