Infinite-dimensional super Lie groups

A super Lie group is a group whose operations are G∞G∞ mappings in the sense of Rogers. Thus the underlying supermanifold possesses an atlas whose transition functions are G∞G∞ 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 hh is a closed, split sub-super Lie algebra of the super Lie algebra of a super Lie group GG, then hh is the super Lie algebra of a sub-super Lie group of GG. Additionally, we show that if gg is a Banach super Lie algebra satisfying certain natural conditions, then there is a super Lie group GG such that the super Lie algebra gg is in fact the super Lie algebra of GG. We also show that if HH is a closed sub-super Lie group of a super Lie group GG, then G→G/HG→G/H is a principal fiber bundle.We emphasize that some of these theorems are known when one works in the super-analytic category and also when the space of supernumbers is finitely generated in which case, one can use finite-dimensional techniques. The issues dealt with here are that our supermanifolds are modeled on graded Banach spaces and that all mappings must be morphisms in the G∞G∞ category.