Lie supergroup

Definition

A finite-dimensional smooth real Lie supergroup is a group object in $\mathbf{SMan}_{\mathbb R}^{\mathrm{sm}}$, the category of smooth real supermanifolds. It consists of a supermanifold $G$ and morphisms

satisfying the associative, inverse, and unit diagrams.

Taking reduced manifolds gives an ordinary real Lie group $G_{\mathrm{red}}$. For every test supermanifold $S$, the set of $S$-points $G(S)$ is a group, naturally in $S$. The ordinary group $G_{\mathrm{red}}$ does not by itself recover the odd directions or their brackets.

The tangent space at the identity carries a canonical Lie superalgebra. In the smooth finite-dimensional category, the full Lie supergroup can equivalently be encoded by a super Harish–Chandra pair.