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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 gijs.tuynman@univ-lille.fr 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 [ Fulltext-pdf (464 KB)] for subscribers only. |