I am looking for a explicit relation for adjoint cohomology of infinite dimensional Lie algebras (specifically smooth vector fields on a manifold), is it possible to apply Hochschild-Serre spectral sequence for these algebras?
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1$\begingroup$ There is no assumption of finite dimensionality for the Hochschild-Serre spectral sequence. $\endgroup$– abxCommented Jan 14, 2018 at 19:36
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$\begingroup$ This is a bit late, but if anybody else comes across this, maybe it's good to address: The original paper "Hochschild, Serre - Cohomology of Lie Algebras" requires in the proof of Theorem 1 the existence of a projection from the Lie algebra $G$ to its subalgebra $K$ (basically, to extend cochains from $K$ to $G$). This may not exist e.g. in the setting of Banach spaces when your subspace is not complemented. Extensions of continuous cochains on a subspace to the whole space seem tricky in infinite dimensions, so there are legitimate concerns IMHO. $\endgroup$– user126256Commented Jun 5, 2020 at 9:46
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A very useful source to learn about cohomology of infinite-dimensional Lie algebras is the book by D.B.Fuks "Cohomology of Infinite-Dimensional Lie Algebras" (shocking, I know). This source discusses Lie algebras of vector fields on manifolds, and applies the Hochschild-Serre spectral sequence in several different contexts. I suspect that going through that would resolve some of your concerns, and maybe settle many more of potential future questions.
answered Jan 15, 2018 at 7:07