Automorphism Supergroups of Supermanifolds

Symmetries of supergeometries related to nonholonomic superdistributions

“…Here α runs through both even and odd indices as well. The following result was originally established in [30,Lemma 13]. We give here a simplified proof that does not use the concept of flow for supermanifolds.…”

“…Then, the forgetful map Aut(Φ) 0 → Aut(Φ 0) is injective with closed image, cf. [30,Lemmas 10 and 11]. It follows from this and Lemma 4.2 that the automorphism supergroup (Aut(Φ) 0, aut(Φ)) is a finite-dimensional super Harish-Chandra pair, in other words a Lie supergroup.…”

“…Our class of geometries is considerably larger, and in addition we consider the infinitesimal symmetry superalgebra that gives a finer dimension bound. We can also vary smoothness in the real case, to which we restrict for simplicity, and our results hold true in the complex analytic or algebraic cases too, as well as in the mixed case considered in [30]. Indeed, our arguments do not rely on any Batchelor realization of M = (M o , A M ) [5], which is well-known to fail for most classes of supermanifolds.…”

“…The symmetries of G-structures are majorized by the tower of principal bundles (3.9) by the classical construction of Sternberg [36], extended to the supercase in [30]: it is proven there that automorphisms of a finite type G-structure F G on a supermanifold M form a Lie supergroup Aut(M, F G ). We will generalize this in what follows.…”

“…The automorphism supergroup in the case of G-structures was studied by Ostermayr [30] though this reference does not contain the supersymmetry dimension bounds. Our class of geometries is considerably larger, and in addition we consider the infinitesimal symmetry superalgebra that gives a finer dimension bound.…”