The Left-Regular Representation of a Super Lie Group

摘要

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.