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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
Full Text:
##### References:
 [1] H.-J. BAUES et D. CONDUCHÉ , The Central Series for Peiffer Commutators in Groups with Operators (J. Algebra, vol. 133, 1990 , pp. 1-34). MR 91m:20051 | Zbl 0704.55008 · Zbl 0704.55008 · doi:10.1016/0021-8693(90)90066-W [2] C. CUVIER , Homologie des algèbres de Leibnitz , C.R. Acad. Sci. Paris, 1991 (à paraître). [3] H. CARTAN et S. EILENBERG , Homological Algebra , Princeton University Press, 1956 . MR 17,1040e | Zbl 0075.24305 · Zbl 0075.24305 [4] C. CHEVALLEY et S. EILENBERG , Cohomology Theory of Lie Groups and Lie Algebras (Trans. of the A.M.S., vol. 63, 1948 , p. 85-124). MR 9,567a | Zbl 0031.24803 · Zbl 0031.24803 · doi:10.2307/1990637 [5] J.-L. LODAY , Cyclic Homology (à paraître). · Zbl 0719.19002 [6] J.-L. LODAY , Opérations sur l’homologie cyclique des algèbres commutatives (Inventiones Mathematicœ, vol. 96, 1990 , p. 205-230). MR 89m:18017 | Zbl 0686.18006 · Zbl 0686.18006 · doi:10.1007/BF01393976 · eudml:143678 [7] J.-L. LODAY et C. PROCESI , Cyclic Homology and Lambda Operations , in K-Theory : connections with Geometry and Topology, NATO ASI series C, vol. 279, 1989 , p. 209-224. MR 91e:19002 | Zbl 0719.19002 · Zbl 0719.19002 [8] J.-L. LODAY et D. QUILLEN , Cyclic Homology and the Lie Algebra Homology of Matrices (Commentarii Math. Helvetici, vol. 59, 1984 , p. 565-591). MR 86i:17003 | Zbl 0565.17006 · Zbl 0565.17006 · doi:10.1007/BF02566367 · eudml:139991 [9] S. MACLANE , Homology , Springer-Verlag, 1963 . MR 28 #122 | Zbl 0133.26502 · Zbl 0133.26502 [10] J. MILNOR et J.-C. MOORE , On the Structure of Hopf Algebras (Annals of Math., vol. 91, 1965 , p. 211-264). MR 30 #4259 | Zbl 0163.28202 · Zbl 0163.28202 · doi:10.2307/1970615
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