Please excuse, very naive question:
Suppose g is a topological Lie algebra over Q and G = exp(g) the associated group
(take free group on formal symbols exp(X), X in G, and impose all relations
formally coming from the BCH-formula).
Suppose I have a short exact sequence
0longrightarrowb1capb2longrightarrowb1oplusb2longrightarrowglongrightarrow0
of g-modules, but special in the sense that
- g is the full Lie algebra,
- b1, b2 (and then their intersection) are supposed to be proper Lie ideals (and not just any kind of g-module)
- and b1+b2=g ; this can happen if g is weird enough
It seems to me (in some cases)/(always)/(never? ;-) such a sequence should induce
something like an exact sequence
1 -> A -> B -> G -> 1
(exact in the obvious classical sense, clearly groups are not an abelian category...)
where B could be something like the coproduct/free product of the normal
subgroups associated to the Lie ideals b1, b2; and A the subgroup associated
to the intersection of b1 and b2.
Is that true or is it complete nonsense? Is it trivially totally ridiculously false?
[please note, even though it may sound so, I do not want to go in
the direction of 'integrating' g-modules to G-modules, I would like to
transfer Lie algebra decompositions to 'nonlinear' group 'decompositions',
whatever that means....]
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