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

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