Abelian extensions and crossed modules of Hom-Lie algebras

  title={Abelian extensions and crossed modules of Hom-Lie algebras},
  author={Jos{\'e} Manuel Casas and Xabier Garc'ia-Mart'inez},
  journal={arXiv: Rings and Algebras},
The exterior product and homology of Hom-Lie algebras
In this article, we use the theory of (non-abelian) exterior product of Hom-Lie algebras to prove the Hopf’s formula for these algebras. As an application, we construct an eight-term sequence in the
Extensions and crossed modules of $n$-Lie Rinehart algebras
We introduce a notion of n-Lie Rinehart algebras as a generalization of Lie Rinehart algebras to n-ary case. This notion is also an algebraic analogue of n-Lie algebroids. We develop representation
On the Capability of Hom-Lie Algebras
A Hom-Lie algebra pL, αLq is said to be capable if there exists a Hom-Lie algebra pH,αH q such that L – H{ZpHq. We obtain a characterisation of capable Hom-Lie algebras involving its epicentre and we
Algebraic exponentiation for Lie algebras.
It is known that the category of Lie algebras over a ring admits algebraic exponents. The aim of this paper is to show that the same is true for the category of internal Lie algebras in an additive,
HNN-extension of involutive multiplicative Hom-Lie algebras
The construction of HNN-extensions of involutive Hom-associative algebras and involutive Hom-Lie algebras is described. Then, as an application of HNN-extension, by using the validity of
Some properties of factor set in regular Hom-Lie algebras
In this paper, we give the definition of isoclinism for regular Hom-Lie algebras and verify some of its properties. Finally, we introduce the factor set and show that the isoclinism and isomorphism
Third Cohomology Group and $$\alpha $$ α -Crossed Extensions of Hom–Lie Algebras
We give a proof of the fact that the third cohomology group $$H^3_\alpha (L,M)$$ H α 3 ( L , M ) of the Hom–Lie algebra $$(L,\alpha _L)$$ ( L , α L ) with coefficients in the Hom- L -module
Biderivations and commuting linear maps on Hom-Lie algebras
The purpose of this paper is to determine skew-symmetric biderivations $\text{Bider}_{\text{s}}(L, V)$ and commuting linear maps $\text{Com}(L, V)$ on a Hom-Lie algebra $(L,\alpha)$ having their
On isoclinism of Hom-Lie algebras


Hom-algebras and homology
Classes of $G$-Hom-associative algebras are constructed as deformations of $G$-associative algebras along algebra endomorphisms. As special cases, we obtain Hom-associative and Hom-Lie algebras as
Notes on 1-parameter formal deformations of Hom-associative and Hom-Lie algebras
Abstract The aim of this paper is to extend to Hom-algebra structures the theory of 1-parameter formal deformations of algebras which was introduced by Gerstenhaber for associative algebras and
On Hom-algebra structures
A Hom-algebra structure is a multiplication on a vector space where the structure is twisted by a homomorphism. The structure of Hom-Lie algebra was introduced by Hartwig, Larsson and Silvestrov and
Representations of Hom-Lie Algebras
In this paper, we study representations of hom-Lie algebras. In particular, the adjoint representation and the trivial representation of hom-Lie algebras are studied in detail. Derivations,
On universal central extensions of Hom-Lie algebras
We develop a theory of universal central extensions of Hom-Lie algebras. Classical results of universal central extensions of Lie algebras cannot be completely extended to Hom-Lie algebras setting,
Monoidal Hom–Hopf Algebras
Hom-structures (Lie algebras, algebras, coalgebras, Hopf algebras) have been investigated in the literature recently. We study Hom-structures from the point of view of monoidal categories; in
Deformations of Lie Algebras using σ-Derivations
A Note on the Categorification of Lie Algebras
In this short note we study Lie algebras in the framework of symmetric monoidal categories. After a brief review of the existing work in this field and a presentation of earlier studied and new