There is a much more general form of the functor that takes a Lie algebra to its envelloping ring. This is not precisely what you asked, but if it would be helpful, try looking at my 1992 thesis, Enveloping rings of universal algebras, University Microfilms, and the paper a bit later in Algebra Universalis. Bill Rowan On Mon, 26 Jan 2009, Tim Porter wrote:
In the relationship between Lie groups and Lie algebras, there is the neat result that the Lie functor L : LieGrp->LieAlg restricts to an equivalence on the simply connected Lie groups, and that the fibre over any given Lie algebra is the category of those G which are isomorphic to quotients of the one simply connected one of them and hence there is a Galois theory interpretation in terms of central extensions.
I am sure that this sort of situation must be much more general than just this case, and is somehow linked to abstract Galois theories. I am hoping that someone can point out really neat categorical results on this (in the literature). I am sure I ought to know them but ...
It does not seem to be in Borceux-Janelidze, and my own library on this area is very sadly thin on the ground.
Thanks in advance,
Tim
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