# The Hochschild-Serre spectral sequence relative to an ideal containing the derived subalgebra

Is the Hochschild-Serre spectral sequence $$H_\bullet(\mathfrak g/\mathfrak h,H_\bullet(\mathfrak h,k))\Rightarrow H_\bullet(\mathfrak g,k)$$ for an extension of Lie algebras $$0\to\mathfrak h\to\mathfrak g\to\mathfrak g/\mathfrak h\to0$$ whose kernel contains the derived subalgebra $[\mathfrak g,\mathfrak g]$ (so that $\mathfrak g/\mathfrak h$ is abelian) special in any way? Does one have extra information on its differentials?

I hope this has been treated in the literature —it seems like an easy case, somehow— but I don't seem to be able to find anything useful.

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Usually it's the opposite extreme which is the easy case; namely, the case when the quotient is semisimple, since then the spectral sequence degenerates at the second page and you get a useful factorisation. I have not seen this case treated, but then it never arose in anything I have done and did not look. – José Figueroa-O'Farrill Jun 10 '11 at 18:22
Well, abelian is like at the other extreme from semisimple, so one can hope that the complexity is in the middle! :) – Mariano Suárez-Alvarez Jun 10 '11 at 18:28