Equivalence of Lie supergroups and super Harish–Chandra pairs

Statement

\left\{ \begin{array}{c} \text{finite-dimensional smooth}\\[-2pt] \text{real Lie supergroups} \end{array} \right\} \simeq \left\{ \begin{array}{c} \text{finite-dimensional smooth real}\\[-2pt] \text{super Harish--Chandra pairs} \end{array} \right\}.

The forward functor sends a Lie supergroup $G$ to

\left(G_{\mathrm{red}},\operatorname{Lie}(G)\right),

with the conjugation action of $G_{\mathrm{red}}$ on its Lie superalgebra.

Conversely, a super Harish–Chandra pair $(G_0,\mathfrak g)$ determines a Lie supergroup whose reduced group is $G_0$ and whose infinitesimal superalgebra is $\mathfrak g$ . One sheaf-level construction uses

\mathcal O_G(U) =\operatorname{Hom}_{U(\mathfrak g_{\bar0})} \bigl(U(\mathfrak g),C^\infty(U)\bigr)

with the compatible $G_0$ -action and Hopf-superalgebra structure.

Exact scope

This theorem concerns the finite-dimensional smooth real Berezin–Leites/Kostant category and even morphisms. Analogous equivalences exist in complex-analytic and algebraic settings under their appropriate hypotheses, but those are distinct statements. The equivalence also explains why a Lie superalgebra by itself does not choose the global topology of the reduced Lie group.

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.
C. Carmeli, L. Caston, and R. Fioresi, Mathematical Foundations of Supersymmetry, EMS, 2011. Publisher record. Relevant: The equivalence theorem in Chapter 7.