Non-Abelian Homological Algebra and Its Applications

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

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“…[12], Theorem 2.39 (ii)). The Lie algebraČ(ε) 2 ⊗ N coincides with its ideal generated by the degenerate elements.…”

Section: Theorem 11 (I) There Is An Isomorphism Of -Modulesmentioning

confidence: 96%

Non-Abelian Homology of Lie Algebras

Inassaridze¹,

Khmaladze²,

Ladra³

2004

Glasgow Math. J.

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“…We give just a brief introduction to these aspects for the category of algebras. A fuller account of cotriple derived functors is given in [3,12]. Note that in some categorical and homological literature the term 'comonad' is used instead of the term 'cotriple'.…”

Section: Preliminaries On Derived Functorsmentioning

confidence: 99%

“…In this section we revisit the Hochschild and cyclic homologies of algebras, using methods of non-abelian homological algebra (we refer here to [4,12,21]). We give several presentations of the cyclic homology of algebras as cotriple derived functors.…”

Section: Cyclic Homology Revisitedmentioning

confidence: 99%

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Cyclic homology via derived functors

Donadze¹,

Inassaridze

Ladra

2010

Homology, Homotopy and Applications

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“…To this end consider the canonical free simplicial resolution of the group G in the category of groups acting on the abelian group A: i i Þ, i ! 0 (see [7]). We will use the equivalence of functors H nþ1 ðÀ; AÞ % L n DerðÀ; AÞ, n !…”

Section: A=½a; a ¼ P;mentioning

confidence: 99%

Higher non-abelian cohomology of groups

Inassaridze¹

2002

Glasgow Math. J.

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Abstract. The first non-abelian cohomology of groups introduced by Guin is extended to any dimensions and for a substantially wider class of coefficients called G-partially crossed P-modules. The first and the second non-abelian cohomologies of groups are described in terms of torsors and extensions of groups respectively. Higher non-abelian cohomology pointed sets are described in terms of cotriple right derived functors of the group of derivations with respect to the first contravariant variable. For any short exact coefficient sequence a long exact cohomology sequence is obtained extending the well-known exact cohomology sequences and higher cohomology of groups with coefficients in any G-group is introduced.2000 Mathematics Subject Classification. 18G50, 18G55. 0. Introduction. Our approach to non-abelian cohomology of groups follows Guin's first non-abelian cohomology [5,6] which differs from the classical first nonabelian cohomology pointed set [10] and from the setting of various papers on nonabelian cohomology [4,2,3] extending the classical exact non-abelian cohomology sequence from lower dimensions [10] to higher dimensions.Guin defined his first non-abelian cohomology group when the coefficient group is a crossed G-module and obtained a six term exact cohomology sequence for any short exact sequence of crossed G-modules.A non-abelian cohomology of groups will be defined in any dimension greater than 0, extending Guin's first non-abelian cohomology group and his exact cohomology sequence to a nine term exact cohomology sequence. A substantially wider class of coefficients will be used, consisting of partially crossed modules over a group P on which a group G acts on the left: these will be called G-partially crossed modules over P. We describe the first non-abelian cohomology in terms of torsors and the second non-abelian cohomology in terms of extensions of groups. Moreover a close relation between non-abelian cohomology of groups and nonabelian right derived functors of the group of derivations will be established and for some particular cases of coefficients a long exact cohomology sequence will be obtained.All considered groups will be arbitrary (not necessarily commutative). An action of a group G on a group A means an action on the left of G on A by automorphisms and will be denoted by g a; g 2 G, a 2 A: We assume that G acts on itself by conjugation. The center of a group G will be denoted by ZðGÞ: If the groups G and R act on a group A then the notation gr a means g ð r aÞ, g 2 G; r 2 R, a 2 A:Glasgow Math.

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Non-Abelian Homological Algebra and Its Applications

Cited by 30 publications

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Non-Abelian Homology of Lie Algebras

Non-Abelian Homology of Lie Algebras

Cyclic homology via derived functors

Higher non-abelian cohomology of groups

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