Kassel, Christian Kähler differentials and coverings of complex simple Lie algebras extended over a commutative algebra. (English) Zbl 0549.17009 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) Cited in 4 ReviewsCited in 59 Documents 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 PDF BibTeX XML Cite \textit{C. Kassel}, J. Pure Appl. Algebra 34, 265--275 (1984; Zbl 0549.17009) Full Text: DOI References: [1] N. Bourbaki, Groupes et Algèbres de Lie (Paris). [2] Cathelineau, J.-L., Sur l’homologie de l’algèbre de Lie de SO(3, \(R\))considérée comme algèbre de Lie sur \(Z\), C.R. acad. sci. Paris, 294, 737-740, (1982) · Zbl 0491.17007 [3] Chevalley, C., Sur certains groupes simples, Tôhoku math. J., 7, 14-66, (1955) · Zbl 0066.01503 [4] Dennis, R.K.; Igusa, K., Hochschild homology and the second obstruction for pseudo-isotopy, (), 7-58 [5] Garland, H., The arithmetic theory of loop groups, Publ. I.H.E.S., 52, 5-136, (1980) · Zbl 0475.17004 [6] van der Kallen, W., Infinitesimally central extensions of Chevalley groups, () · Zbl 0275.17006 [7] Kassel, C., Calcul algébrique de l’homologie de certains groupes de matrices, J. algebra, 80, 235-260, (1983) · Zbl 0511.18014 [8] Kassel, C.; Loday, J.-L., Extensions centrales d’algèbres de Lie, Ann. inst. Fourier, 32, 119-142, (1982) · Zbl 0485.17006 [9] Milnor, J., Introduction to algebraic K-theory, () · Zbl 0237.18005 [10] Milnor, J., On the homology of Lie groups made discrete, Comment. math. helv., 58, 72-85, (1983) · Zbl 0528.20033 [11] Springer, T.A.; Steinberg, R., Conjugacy classes, (), 167-266 · Zbl 0249.20024 [12] Steinberg, R., Générateurs, relations et revêtements de groupes algébriques, Colloque sur la théorie des groupes algébriques, 113-127, (1962), Bruxelles · Zbl 0272.20036 [13] Suslin, A.A., On the K-theory of local fields, (1983), Preprint · Zbl 0525.18008 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.