Friday, 19 June 2009

lie algebras - In what way do exact sequences of Lie ideals integrate to the category of groups?

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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