Lie superalgebra of a Lie supergroup

Core idea

Let $G$ be a Lie supergroup with identity $e$ . Its Lie superalgebra is the tangent super vector space

\operatorname{Lie}(G)=T_eG.

Left translation identifies $T_eG$ with the left-invariant vector fields on $G$ . Transporting the supercommutator of derivations through this identification gives a Lie superbracket on $T_eG$ .

The even part is canonically the ordinary Lie algebra of the reduced group:

\operatorname{Lie}(G)_{\bar 0} \cong \operatorname{Lie}(G_{\mathrm{red}}).

The odd tangent space supplies the genuinely new infinitesimal directions, and its symmetric odd–odd bracket can have values in the even Lie algebra.

Functoriality

A homomorphism of Lie supergroups $F:G\to H$ differentiates at the identity to an even Lie-superalgebra homomorphism

dF_e:\operatorname{Lie}(G)\to\operatorname{Lie}(H).

This defines a Lie functor from smooth real Lie supergroups to finite-dimensional real Lie superalgebras. Unlike the ordinary simply-connected integration statement, a Lie superalgebra alone does not record the chosen global reduced Lie group; the missing global data is organized by a super Harish–Chandra pair.

References

B. Kostant, “Graded manifolds, graded Lie theory, and prequantization,” in Differential Geometrical Methods in Mathematical Physics, Lecture Notes in Mathematics 570, Springer, 1977, 177–306. Chapter.
V. S. Varadarajan, Supersymmetry for Mathematicians: An Introduction, Courant Lecture Notes 11, American Mathematical Society, 2004. Publisher record. Relevant: Chapter 7.