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Central extensions of Lie algebras. (Extensions centrales d’algèbres de Lie.) (French) Zbl 0485.17006

Authors’ summary: Given a commutative ring \(k\) and an associative \(k\)-algebra \(A\), we compute the homology group \(H_2 (\mathfrak{sl}_n(A), k)\) of the \(k\)-Lie algebra \(\mathfrak{sl}_n(A)\) of “trace zero” matrices. This group appears to be a homology group of a complex derived from A. Connes’ work; it is isomorphic to \(\Omega^1_{A/k}/dA\) when \(A\) is commutative. Results are also given for relative homology associated to a surjection of \(k\)-algebras. The proofs involve a classification of central extensions and crossed modules of Lie algebras.

MSC:

17B55 Homological methods in Lie (super)algebras
18F25 Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects)
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