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Leibniz algebras: Definitions, properties. (Algèbres de Leibniz: Définitions, propriétés.) (French) Zbl 0821.17024
This is a long paper covering many aspects of Leibniz algebras. The Leibniz algebra is a noncommutative version of Lie algebra, which we get when omitting the condition \([x,x] = 0\) and keeping the Jacobi identity in a suitable form. The paper is divided into three chapters.
In the first chapter the author defines Leibniz algebras, introduces the center and essential center of a Leibniz algebra, defines a module over a Leibniz algebra, and determines all 2-dimensional Leibniz algebras over a field. Then we can find here a description of Leibniz algebras with low dimensional or low codimensional center (mostly the dimension or codimension 1 is treated). Finally, he discusses the relation of Leibniz algebras to partial Lie algebras as introduced by H. J. Baues and D. Conduché [J. Algebra 133, 1-34 (1990; Zbl 0704.55008)].
The second chapter is devoted to the homology and cohomology of a Leibniz algebra \(L\) with coefficients in a module \(M\) over this algebra. The author first introduces these quite nontrivial notions (comparing them with their Lie algebra counterparts) and computes some examples using his classification of 2-dimensional Leibniz algebras. He determines homology and cohomology of Leibniz algebras in dimensions 0 and 1, and uses second and third cohomology groups for the description of abelian extensions of Leibniz algebras and the description of crossed modules over a Leibniz algebra. He also introduces and studies a cup product in the cohomology of Leibniz algebras.
The most important results can be found in the last third chapter of the paper. For an associative and unital algebra \(A\) over a field \(k\) of characteristic zero let \({\mathfrak g\mathfrak l}(A)\) denote the Lie algebra of finite matrices with entries from \(A\) considered as a Leibniz algebra. Constructing an auxiliary complex the author shows that \[ HL_ * ({\mathfrak g\mathfrak l} (A); k) \cong T^* (HH_ * (A) [-1]). \] Here \(HL_ *\) denotes the homology functor on the category of Leibniz algebras, \(HH_ *\) denotes the Hochschild homology, and \(T^*\) denotes the tensor algebra functor. This is a noncommutative version of the Loday-Quillen theorem giving the isomorphism \[ H_ * ({\mathfrak g \mathfrak l} (A); k) \cong \Lambda^* HC_{* - 1} (A) \] J. L. Loday and D. Quillen [Comment. Math. Helv. 59, 565-591 (1984; Zbl 0565.17006)]. The author also presents a result about the stability of \(HL_ * ({\mathfrak g\mathfrak l}_ n(A))\) and introduces \(\lambda\)-operations.
Reviewer: J.Vanžura (Brno)

MSC:
17A32 Leibniz algebras
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References:
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