Abelian extensions and crossed modules of Hom-Lie algebras

2021

Crossed modules enjoyed a lively research activity in the late 1940s after Whitehead introduced them to classify 2-types [Whitehead, 1949]. Said activity was arguably revived after Loday [Loday, 1982] justified the classification of all types using the so-called n-cat Groups. This 1990s revival, spearheaded by [Norrie, 1987], aimed at studying crossed modules as algebraic objects in their own right. This paper is dedicated to crossed modules having a smooth structure, thought of as the global counterpart to Lie 2-algebras [Baez-Crans, 2004]. Recall the definition of a Crossed module (e.g., [Baez-Lauda, 2004]).

Algebra Colloquium, 2000

Extensions of crossed modules in Lie algebras with abelian kernel are studied, particularly backward and forward induced extensions and related properties. The set Opext((U, Q, ε), (R, K, ∂)) of congruence classes of extensions of (R, K, ∂) by (U, Q, ε) is endowed with a K-vector space structure. This Kvector space appears in a five-term natural and exact sequence associated with an extension of crossed modules.

2019

Let $ 0 \rightarrow A\rightarrow L {\rightarrow} B \rightarrow 0 $ be an abelian extension of Lie algebras. In this paper, we construct certain exact sequences which relate derivations with the Lie algebra cohomology group $ H^{2}(B,A) $, and apply them to study extending derivations of $ A $ and lifting derivations of $ B $ to certain derivations of $ L $.

Glasgow Mathematical Journal

The purpose of this paper is to define an α-type cohomology, which we call α-type Chevalley–Eilenberg cohomology, for Hom-Lie algebras. We relate it to the known Chevalley–Eilenberg cohomology and provide explicit computations for some examples. Moreover, using this cohomology, we study formal deformations of Hom-Lie algebras, where the bracket as well as the structure map α are deformed. Furthermore, we provide a generalization of the grand crochet and study, in a particular case, the α-type cohomology for Hom-Lie bialgebras.

Glasgow Mathematical Journal, 2004

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 relation of cyclic homology with Milnor cyclic homology of non-commutative associative algebras is established in terms of the long exact non-abelian homology sequence of Lie algebras. Some explicit formulas for the second and the third non-abelian homology of Lie algebras are obtained. Using the generalised notion of the Lie algebra of derivations, we introduce the second non-abelian cohomology of Lie algebras with coefficients in crossed modules and extend the seven-term exact non-abelian cohomology sequence of Guin to nine-term exact sequence.

Journal of Geometry and Physics, 2011

The aim of this paper is to provide cohomologies of n-ary Hom-Nambu-Lie algebras governing central extensions and one parameter formal deformations. We generalize to n-ary algebras the notions of derivations and representation introduced by Sheng for Hom-Lie algebras. Also we show that a cohomology of n-ary Hom-Nambu-Lie algebras could be derived from the cohomology of Hom-Leibniz algebras.

Mediterranean Journal of Mathematics, 2017

In this paper we investigate in details derivations on trivial extension algebras. We obtain generalizations of both known results on derivations on triangular matrix algebras and a known result on first cohomology group of trivial extension algebras. As a consequence we get the characterization of trivial extension algebras on which every derivation is inner. We show that, under some conditions, a trivial extension algebra on which every derivation is inner has necessarily a triangular matrix representation. The paper starts with detailed study (with examples) of the relation between the trivial extension algebras and the triangular matrix algebras.

Arxiv preprint arXiv:1108.2806, 2011

In this paper we aim to understand the category of stable-Yetter-Drinfeld modules over enveloping algebra of Lie algebras. To do so, we need to define such modules over Lie algebras. These two categories are shown to be isomorphic. A mixed complex is defined for a given Lie algebra and a stable-Yetter-Drinfeld module over it. This complex is quasi-isomorphic to the Hopf cyclic complex of the enveloping algebra of the Lie algebra with coefficients in the corresponding module. It is shown that the (truncated) Weil algebra, the Weil algebra with generalized coefficients defined by Alekseev-Meinrenken, and the perturbed Koszul complex introduced by Kumar-Vergne are examples of such a mixed complex.

2000

If L is a Lie algebra over R and Z its centre, the natural inclu- sion Z, ! (L) extends to a representation i :Z! EndH(L;R )o f the exterior algebra of Z in the cohomology of L. We begin a study of this rep- resentation by examining its Poincar e duality properties, its associated higher cohomology operations and its

arXiv: Differential Geometry, 2018

In this thesis, we introduce a new cohomology theory associated to a Lie 2-algebras and a new cohomology theory associated to a Lie 2-group. These cohomology theories are shown to extend the classical cohomology theories of Lie algebras and Lie groups in that their second groups classify extensions. We use this fact together with an adapted van Est map to prove the integrability of Lie 2-algebras anew.