Xabier García‐Martínez

activity

20142026

most citedNon-abelian tensor product and homology of Lie Superalgebras

30 citations · 84 across the 15 of their papers we have counts for

collaborators

17 papers

math.RT2026

Representations of compatible Lie algebras

Xabier García-Martínez, Manuel Ladra, Bernardo Leite da Cunha +1

We study compatible Lie algebras from algebraic and representation-theoretic points of view, obtaining counterexamples to some fundamental theorems from classical Lie algebra theor…

math.RA2026

Non-isomorphic restricted Lie algebras with isomorphic restricted enveloping algebras

Xabier García-Martínez

Let $p$ be a prime. For every field $F$ of characteristic $p$ we exhibit pairs of non-isomorphic finite-dimensional $p$ -nilpotent restricted Lie algebras $L$ and $H$ over $F$ whose…

math.RA2022★ 1 cited

A characterisation of Lie algebras using ideals and subalgebras

Vladimir Dotsenko, Xabier García-Martínez

We prove that if, for a nontrivial variety of non-associative algebras, every subalgebra of every free algebra is free and $I^{2}$ is an ideal whenever $I$ is an ideal, then this var…

math.RA2021

On some properties of $Lie$ -centroids of Leibniz algebras

José Manuel Casas, Xabier García-Martínez, Natalia Pachego-Rego

We study some properties on $Lie$ -centroids related to central $Lie$ -derivations, generalized $Lie$ -derivations and almost inner $Lie$ -derivatio…

math.RA2021

Actor of a crossed module of dialgebras via tetramultipliers

José Manuel Casas, Rafael Fernández-Casado, Xabier García-Martínez +1

We study the representability of actions in the category of crossed modules of dialgebras via tetramultipliers. We deduce a pair of dialgebras in order to construct an object which…

math.RA2021

On the capablility of Hom-Lie algebras

José Manuel Casas, Xabier García-Martínez

A Hom-Lie algebra $(L, α_{L})$ is said to be capable if there exists a Hom-Lie algebra $(H, α_{H})$ such that $L ≅ H / Z (H)$ . We obtain a characterisation of capable Hom-Lie algebras…

math.GR2020

Some generalisations of Schur's and Baer's theorem and their connection with homological algebra

Guram Donadze, Xabier García-Martínez

Schur's Theorem and its generalisation, Baer's Theorem, are distinguished results in group theory, connecting the upper central quotients with the lower central series. The aim of…

math.CT2020★ 2 cited

Algebraic exponentiation for Lie algebras

Xabier García-Martínez, James R. A. Gray

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 algebr…

math.RA2018

Abelian extensions and crossed modules of Hom-Lie algebras

José-Manuel Casas, Xabier García-Martínez

In this paper we study the low dimensional cohomology groups of Hom-Lie algebras and their relation with derivations, abelian extensions and crossed modules. On one hand, we introd…

math.CT2017★ 19 cited

A characterisation of Lie algebras via algebraic exponentiation

Xabier García-Martínez, Tim Van der Linden

In this article we describe varieties of Lie algebras via algebraic exponentiation, a concept introduced by Gray in his Ph.D. thesis. For $K$ an infinite field of characte…

math.RA2017★ 20 cited

A characterisation of Lie algebras amongst anti-commutative algebras

Xabier García-Martínez, Tim Van der Linden

Let $K$ be an infinite field. We prove that if a variety of anti-commutative $K$ -algebras - not necessarily associative, where $xx = 0$ is an identity - is locally…

math.RA2017★ 7 cited

A note on split extensions of bialgebras

Xabier García-Martínez, Tim Van der Linden

We prove a universal characterization of Hopf algebras among cocommutative bialgebras over a field: a cocommutative bialgebra is a Hopf algebra precisely when every split extension…

math.RA2016★ 1 cited

Actor of a crossed module of Leibniz algebras

José Manuel Casas, Rafael F. Casado, Xabier García-Martínez +1

We extend to the category of crossed modules of Leibniz algebras the notion of biderivation via the action of a Leibniz algebra. This results into a pair of Leibniz algebras which…

math.RA2016

Universal central extensions of superdialgebras of matrices

Xabier García-Martínez, Manuel Ladra

We complete the problem of finding the universal central extension in the category of Leibniz superalgebras of $sl (m, n, D)$ when $m + n \geq 3$ and $D$ is a superdialgebr…

math.RA2014★ 30 cited

Non-abelian tensor product and homology of Lie Superalgebras

Xabier García-Martínez, Emzar Khmaladze, Manuel Ladra

We introduce the non-abelian tensor product of Lie superalgebras, study some of its properties including nilpotency, solvability and Engel, and we use it to describe the universal…

math.RT2014★ 3 cited

Universal central extensions of $sl (m, n, A)$ of small rank over associative superalgebras

Xabier García-Martínez, Manuel Ladra

We complete the solution of the problem of finding the universal central extension of the matrix superalgebras $sl (m, n, A)$ where $A$ is an associative superalgebra and…

math.RA2014★ 1 cited

Universal central extensions of Lie-Rinehart algebras

José Luis Castiglioni, Xabier García-Martínez, Manuel Ladra

In this paper we study the universal central extension of a Lie--Rinehart algebra and we give a description of it. Then we study the lifting of automorphisms and derivations to cen…

Representations of compatible Lie algebras

Non-isomorphic restricted Lie algebras with isomorphic restricted enveloping algebras

A characterisation of Lie algebras using ideals and subalgebras

On some properties of Lie-centroids of Leibniz algebras

Actor of a crossed module of dialgebras via tetramultipliers

On the capablility of Hom-Lie algebras

Some generalisations of Schur's and Baer's theorem and their connection with homological algebra

Algebraic exponentiation for Lie algebras

Abelian extensions and crossed modules of Hom-Lie algebras

A characterisation of Lie algebras via algebraic exponentiation

A characterisation of Lie algebras amongst anti-commutative algebras

A note on split extensions of bialgebras

Actor of a crossed module of Leibniz algebras

Universal central extensions of superdialgebras of matrices

Non-abelian tensor product and homology of Lie Superalgebras

Universal central extensions of sl(m,n,A) of small rank over associative superalgebras

Universal central extensions of Lie-Rinehart algebras

On some properties of $Lie$ -centroids of Leibniz algebras

Universal central extensions of $sl (m, n, A)$ of small rank over associative superalgebras