Non-Abelian Homology of Lie Algebras

Abstract

Abstract. Non-abelian homology of Lie algebras with coefficients in Lie algebras is constructed and studied, generalising the classical Chevalley-Eilenberg homology of Lie algebras. The relationship between cyclic homology and Milnor cyclic homology of non-commutative associative algebras is established in terms of the long exact nonabelian homology sequence of Lie algebras. Some explicit formulae for the second and the third non-abelian homology of Lie algebras are obtained.2000 Mathematics Subject Classification. 17B40, 17B56, 18G10, 18G50. 0. Introduction. The non-abelian homology of groups with coefficients in groups was constructed and investigated in [16,17], using the non-abelian tensor product of groups of Brown and Loday [4, 5] and its non-abelian left derived functors. It generalises the classical Eilenberg-MacLane homology of groups and extends Guin's low dimensional non-abelian homology of groups with coefficients in crossed modules [9], having an interesting application to the algebraic K-theory of non-commutative local rings [9,17].The purpose of this paper is to set up a similar non-abelian homology theory for Lie algebras and is mainly dedicated to state and prove several desirable properties of this homology theory.In [8] Ellis introduced and studied the non-abelian tensor product of Lie algebras which is a Lie structural and purely algebraic analogue of the non-abelian tensor product of groups of Brown and Loday [4,5], arising in applications to homotopy theory of a generalised Van Kampen theorem.Applying this tensor product of Lie algebras, Guin defined the low-dimensional non-abelian homology of Lie algebras with coefficients in crossed modules [10].We construct a non-abelian homology H * (M, N) of a Lie algebra M with coefficients in a Lie algebra N as the non-abelian left derived functors of the tensor product of Lie algebras, generalising the classical Chevalley-Eilenberg homology of Lie algebras and extending Guin's non-abelian homology of Lie algebras [10]. We give an application of our long exact homology sequence to cyclic homology of associative algebras, correcting the result of [10]. In fact, for a unital associative (non-commutative) algebra A we obtain a long exact non-abelian homology available at https://www.cambridge.org/core/terms. https://doi

Homology and central extensions of Leibniz and Lie $n$-algebras

“…For n = 2 the Lie algebra U in Construction 5.3 is isomorphic to the non-abelian Lie exterior square of L. Thus if L is a Vect-perfect Lie algebra, then it is the same as the non-abelian Lie tensor square of L [13]. This fact follows easily from [18,Proposition 1]. Moreover, we recover the description of the universal central extension of a Vect-perfect Lie algebra obtained in [13,Theorem 11].…”

A Non-abelian Tensor Product of Hom–Lie Algebras

“…Since the actions of (M, α M ) and (N , α N ) on each other are compatible, it follows by direct calculations that the product (2) satisfies the skew-symmetry and the Hom-Jacobi identity. [6] (see also [7,11]). …”

“…Of course for α A = id A , this is the definition of the first Milnor cyclic homology of the associative algebra A in the sense of [13] (see also [11] …”

Peiffer elements in simplicial groups and algebras

“…The decomposition of the group of boundaries of the Moore complex of a simplicial group (algebra) as a product of commutator subgroups (sum of product ideals) is of interest in various topological and homological settings. For instance, in calculations of non-abelian homology of groups [12] and non-abelian homology of Lie algebras [13], and to explain the relations among several algebraic models of connected homotopy 3-types: braided regular crossed modules, 2-crossed modules, quadratic modules, crossed squares and simplicial groups with Moore complex of length 2 [1,3,15,16]. It is possible that such decomposition contributes to light complete descriptions of algebraic models of the n-types of specific families of spaces for low values of n, to calculate Samelson and Whitehead products [10] and analogues in homotopy theory of simplicial Lie algebras or to link simplicial groups and weak infinity categoric models.…”