## 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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### References:

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