Spectral sequence in Lie algebra

Question

The exact sequence $0\rightarrow sl(A)\rightarrow gl(A)\rightarrow A/[A,A]\rightarrow 0$ gives rise to a spectral sequence in homology, I want to know the details for this spectral sequence at the second page?

Answer 1

This is explained in the article of Beno Eckmann and Urs Stammbach "On exact sequences in the homology of groups and algebras". The Hochschild-Serre spectral sequence for homology of Lie algebras, applied to the short exact sequence $0\rightarrow N \rightarrow G \rightarrow Q \rightarrow 0$ of Lie algebras, and every $Q$-module $M$ yields long exact sequences in homology, see $(I)$ to $(VIII)$ in Eckmann's and Stammbach's paper: $(I)$ is $H_2(G,M)\rightarrow H_2(Q,M)\rightarrow \cdots H_1(G,M)\rightarrow H_1(Q,M)\rightarrow 0$, $(VIII)$ is $H_n(G,M)\rightarrow H_n(Q,M)\rightarrow H_{n-2}(Q,H_1(N,M))\rightarrow \cdots \rightarrow H_1(Q,M)\rightarrow 0$.