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.


17B55 Homological methods in Lie (super)algebras
18F25 Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects)
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[1] S. BLOCH, The dilogarithm and extensions of Lie algebras, Alg. K-theory, Evanston 1980, Springer Lecture Notes in Math., n° 854 (1981), 1-23. · Zbl 0469.14009
[2] H. CARTAN and S. EILENBERG, Homological algebra, Princeton University Press (1956). · Zbl 0075.24305
[3] H. GARLAND, The arithmetic theory of loop groups, Publ. I.H.E.S., n° 52 (1980), 5-136. · Zbl 0475.17004
[4] D. GUIN-WALERY et J.-L. LODAY, Obstruction à l’excision en K-théorie algébrique, Alg. K-theory, Evanston 1980, Springer Lect. Notes in Math., n° 854 (1981), 179-216. · Zbl 0461.18007
[5] G. HOCHSCHILD, Lie algebra kernels and cohomology, Amer. J. Math., 76 (1954), 698-716. · Zbl 0055.26601
[6] C. KASSEL, Homologie du groupe linéaire général et K-théorie stable, thèse, Université de Strasbourg, juin 1981. · Zbl 0445.20020
[7] C. KASSEL, Calcul algébrique de l’homologie de certains groupes de matrices, J. of Algebra, 80, n° 1 (1983). · Zbl 0511.18014
[8] J.-L. LODAY, Cohomologie et groupe de Steinberg relatifs, J. of Algebra, 54 (1978), 178-202. · Zbl 0391.20040
[9] J. MILNOR, Introduction to algebraic K-theory, Ann. of Math. Studies, n° 72, Princeton University Press (1971). · Zbl 0237.18005
[10] M. MORI, On the three dimensional cohomology group of Lie algebras, J. Math. Soc. Japan, 5 (1953), 171-183. · Zbl 0051.02304
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