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I have to work with Lie algebras of some infinite dimensional 'Lie groups' (e.g. $\Omega SL_2(\mathbb{C})$) but i'm not sure on how to approach infinite dimensional groups, for loop group it is not so obvious what should be considered a neighborhood of the identity. I don't want to see the whole solution, maybe just some hints, but much more i would appreciate an explanation (or some reference) of how one should view local structure of such groups.

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"As in finite dimensions, the tangent space $L(G) := T_1(G)$ at the identity element of a Lie group $G$ can be made a topological Lie algebra via the identification with the Lie algebra of left invariant vector fields on $G$."

Reference: Helge Glöckner, Fundamental Problems in the Theory of Infinite-Dimensional Lie groups.

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