3 - Super Lie groups. General theory

Summary

Definition and structure of super Lie groups

Definition

We first recall the definition of a group. A group is a set G endowed with a binary-operation mapping F:G × G→G, abbreviated by F(x, y) = xy for all x, y in G and called multiplication, which has the following properties:

1. (1) (xy)z = x(yz) = xyz, for all x, y, z in G.

2. (2) There exists an element e of G, called the identity, such that ex = xe = x for all x in G.

3. (3) For every x in G there exists an element of G called the inverse of x, written x^-1, such that x^-1x = xx^-1 = e.

It is easy to verify that e is a unique element of G and that every element has a unique inverse. In particular, e^-1 = e and (x^-1)^-1 = x for all x in G.

A super Lie group is a group G that has also the following additional properties:

1. (4) It is a supermanifold, the points of which are the group elements.

2. (5) The multiplication mapping F is differentiate.

If, in the last two sentences, one replaces ‘supermanifold’ by ‘manifold’ and ‘differentiable’ by ‘C^∞’ one has the definition of an ordinary Lie group.

Associated with every super Lie group G is an infinite family of ordinary Lie groups, each of which is obtained by replacing G with one of its skeletons (see section 2.1).