Kähler differentials and coverings of complex simple Lie algebras extended over a commutative algebra. (English) Zbl 0549.17009

Kassel, Christian

J. Pure Appl. Algebra 34, 265-275 (1984).

Let \(k\) be a commutative ring with unit, \(A\) an associative \(k\)-algebra, \(P\) an \(A\)-bimodule, \(M_ n(P)\) the \(k\)-module of \(n\times n\) matrices with entries in \(P\). The following theorem is presented. If \(A\) and \(P\) are free \(k\)-modules then for \(n\geq 5\) the groups \(H_ 1(\mathfrak{sl}_ n(A),M_ n(P))\) and \(H_ 1(\mathfrak{sl}_ n(A),M_ n'(P))\) are both isomorphic to the Hochschild homology group \(H_ 1(A,P)\). A theory of central extensions in the category of Lie algebra modules is sketched, and the results are interpreted in terms of central extensions. The last section deals with the remaining types of finite-dimensional complex simple Lie algebras.

Reviewer: Alice Fialowski (Budapest)

MSC:

17B55	Homological methods in Lie (super)algebras
17B20	Simple, semisimple, reductive (super)algebras
17B56	Cohomology of Lie (super)algebras

Keywords:

Schur multiplier; covering; Hochschild homology group; central extensions; complex simple Lie algebras

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