Gnedbaye, Allahtan V. A non-abelian tensor product of Leibniz algebras. (English) Zbl 0936.17004 Ann. Inst. Fourier 49, No. 4, 1149-1177 (1999). Let \({\mathfrak M}\) and \({\mathfrak N}\) be crossed Lie \({\mathfrak g}\)-algebras. D. Guin [Ann. Inst. Fourier 45, No. 1, 93-118 (1995; Zbl 0818.17022)] used the derivation functor \(\text{Der}_{\mathfrak g}({\mathfrak N},-)\) and its right adjoint, the non-abelian tensor product \(-\otimes_{\mathfrak g}{\mathfrak N}\) to construct a non-abelian (co)homology theory. The aim of the paper is to extend these methods and results on Leibniz algebras – a non-commutative variation of Lie algebras. The author compares the Milnor-type Hochschild homology \(HH^M_\ast(A)\) and the classical Hochschild homology \(HH_\ast(A)\) of an associative algebra \(A\). Reviewer: M.Golasiński (Toruń) Cited in 1 ReviewCited in 11 Documents MSC: 17A32 Leibniz algebras 18G50 Nonabelian homological algebra (category-theoretic aspects) 16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.) 17B55 Homological methods in Lie (super)algebras 17B56 Cohomology of Lie (super)algebras Keywords:Milnor-type Hochschild homology; crossed module; Leibniz algebra; non-abelian Leibniz (co)homology; non-abelian tensor product PDF BibTeX XML Cite \textit{A. V. Gnedbaye}, Ann. Inst. Fourier 49, No. 4, 1149--1177 (1999; Zbl 0936.17004) Full Text: DOI Numdam EuDML References: [1] J.-M. CASAS & M. LADRA, Perfect crossed modules in Lie algebras, Comm. Alg., 23(5) (1995), 1625-1644. · Zbl 0860.17032 [2] Ch. CUVIER, Algèbres de leibnitz: définitions, propriétés, Ann. Ecole Norm. Sup., (4) 27 (1994), 1-45. · Zbl 0821.17024 [3] G.J. ELLIS, A non-abelian tensor product of Lie algebras, Glasgow Math. J., 33 (1991), 101-120. · Zbl 0724.17016 [4] A.V. GNEDBAYE, Third homology groups of universal central extensions of a Lie algebra, Afrika Matematika (to appear), Série 3, 10 (1998). · Zbl 1054.17003 [5] D. GUIN, Cohomologie des algèbres de Lie croisées et K-théorie de Milnor additive, Ann. Inst. Fourier, Grenoble, 45-1 (1995), 93-118. · Zbl 0818.17022 [6] J.-L. LODAY, Cyclic homology, Grund. math. Wiss., Springer-Verlag, 301, 1992. · Zbl 0780.18009 [7] J.-L. LODAY, Une version non commutative des algèbres de Lie: les algèbres de Leibniz, L’Enseignement Math., 39 (1993), 269-293. · Zbl 0806.55009 [8] J.-L. LODAY & T. PIRASHVILI, Universal enveloping algebras of Leibniz algebras and (co)homology, Math. Annal., 296 (1993), 139-158. · Zbl 0821.17022 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.