In this paper, we extend the theory of super Harish-Chandra pairs, originally developed by Kostant and Koszul for smooth Lie supergroups, to algebraic supergroups over a field of characteristic zero. We also review the corresponding complex analytic theory and we give a characterization of the action of an algebraic (resp. complex analytic) super Harish--Chandra pair on a supervariety (resp. complex analytic supermanifold).
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