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Journal of Lie Theory 29 (2019), No. 1, 001--078
Copyright Heldermann Verlag 2019

The Left-Regular Representation of a Super Lie Group

Gijs M. Tuynman
Dép. de Mathématiques, Faculté des Sciences et Technologies, Université de Lille, 59655 Villeneuve d'Ascq, France

With the usual definition of a super Hilbert space and a super unitary representation, it is easy to show that there are lots of super Lie groups for which the left-regular representation is not super unitary. I will show that weakening the definition of a super Hilbert space (by allowing the super scalar product to be non-homogeneous, not just even) will allow the left-regular representation of all (connected) super Lie groups to be super unitary (with an adapted definition). Along the way I will introduce a (super) metric on a supermanifold that will allow me to define super and non-super scalar products on function spaces.

Keywords: Super manifolds, super Lie groups, regular representation, super unitary representation.

MSC: 58A50, 22E99; 57S20

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