Coproduct and cogroups in the category of graded dual Leibniz algebras. (English) Zbl 0880.17002

Oudom, Jean-Michel

Loday, Jean-Louis (ed.) et al., Operads: Proceedings of renaissance conferences. Special session and international conference on moduli spaces, operads, and representation theory/operads and homotopy algebra, March 1995/May–June 1995, Hartford, CT, USA/Luminy, France. Providence, RI: American Mathematical Society. Contemp. Math. 202, 115-135 (1997).

J.-L. Loday [Cyclic homology, Springer Verlag, Grundlehren Math. Wiss. 301 (1992; Zbl 0780.18009)] defined Leibniz algebras as some generalization of Lie algebras (there, we assume the Jacobi identity but not antisymmetry) and built the so-called Leibniz homology \(HL_*(\cdot)\) of these objects. Some fault in this theory is that the classical Künneth isomorphism \(HL_*({\mathfrak g}'\times{\mathfrak g}'')\cong HL_*({\mathfrak g}')* HL_*({\mathfrak g}'')\) and the isomorphism \(HL_*(gl(A))\cong THH_{*-1}(A)\) for the homology of Lie algebras of matrices with coefficients in an associative algebra \(A\) with unit are valid only in the category of graded vector spaces, not in the category of algebras [as we have for the Lie homology].
The aim of this paper is a preferable comprehension of the above parallelism between Lie homology and Leibniz homology. Namely, the author exhibits some algebraic structures (on the basis of dual Leibniz algebras) for which the isomorphisms above would be isomorphisms of coalgebras or algebras. Finally, he gives a new proof of the Cuvier-Loday theorem computing \(HL_*(gl(A))\) in terms of cogroups in the category of graded dual Leibniz algebras: \(HL^*(gl(A))\cong THH^{*-1}(A)\).
For the entire collection see [Zbl 0855.00018].

Reviewer: J.Kubarski (Łódź)

MSC:

17A30	Nonassociative algebras satisfying other identities
17B55	Homological methods in Lie (super)algebras
17B56	Cohomology of Lie (super)algebras
18D35	Structured objects in a category (MSC2010)

Keywords:

Künneth formula; comultiplicative object; Leibniz algebras; Leibniz homology; Lie algebras of matrices; dual Leibniz algebras; coalgebras; Cuvier-Loday theorem

Citations:

Zbl 0780.18009