Infinite dimensional super Lie groups

Abstract

A super Lie group is a group whose operations are $G^{\infty}$ mappings in the sense of Rogers. Thus the underlying supermanifold possesses an atlas whose transition functions are $G^{\infty}$ functions. Moreover the images of our charts are open subsets of a graded infinite-dimensional Banach space since our space of supernumbers is a Banach Grassmann algebra with a countably infinite set of generators. In this context, we prove that if $\hfrak$ is a closed, split sub-super Lie algebra of the super Lie algebra of a super Lie group $\Gcal,$ then $\hfrak$ is the super Lie algebra of a sub-super Lie group of $\Gcal.$ Additionally, we show that if $\gfrak$ is a Banach super Lie algebra satisfying certain natural conditions, then there is a super Lie group $\Gcal$ such that the even part of $\gfrak$ is the even part of the super Lie algebra of $\Gcal.$ In general, the module structure on $\gfrak$ is required to obtain $\Gcal,$ but the "structure constants" involving the odd part of $\gfrak$ can not be recovered without further restrictions. We also show that if $\Hcal$ is a closed sub-super Lie group of a super Lie group $\Gcal,$ then $\Gcal \rar \Gcal/\Hcal$ is a principal fiber bundle. Finally, we show that if $\gfrak$ is a graded Lie algebra over $C,$ then there is a super Lie group whose super Lie algebra is the Grassmann shell of $\gfrak.$ We also briefly relate our theory to techniques used in the physics literature.