Xabier García-Martínez

UNIVERSAL CENTRAL EXTENSIONS OF LIE-RINEHART ALGEBRAS

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 central extensions. We also give a definition of a non-abelian tensor product in Lie–Rinehart algebras based on the construction of Ellis of non-abelian tensor product of Lie algebras. We relate this non-abelian tensor product to the universal central extension.

Non-abelian tensor product and homology of Lie superalgebras

Abstract We introduce the non-abelian tensor product of Lie superalgebras and study some of its properties. We use it to describe the universal central extensions of Lie superalgebras. We present the low-dimensional non-abelian homology of Lie superalgebras and establish its relationship with the cyclic homology of associative superalgebras. We also define the non-abelian exterior product and give an analogue of Millers theorem, Hopf formula and a six-term exact sequence for the homology of Lie superalgebras.

A note on split extensions of bialgebras

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 over it admits a join decomposition. We also explain why this result cannot be extended to a non-cocommutative setting.

A New Characterisation of Groups Amongst Monoids

We prove that a monoid M is a group if and only if, in the category of monoids, all points over M are strong. This sharpens and greatly simplifies a result of Montoli, Rodelo and Van der Linden (Pré-Publicações DMUC 16–21, 1–41 2016) which characterises groups amongst monoids as the protomodular objects.

A Natural Extension of the Universal Enveloping Algebra Functor to Crossed Modules of Leibniz Algebras

The universal enveloping algebra functor between Leibniz and associative algebras defined by Loday and Pirashvili is extended to crossed modules. We prove that the universal enveloping crossed module of algebras of a crossed module of Leibniz algebras is its natural generalization. Then we construct an isomorphism between the category of representations of a Leibniz crossed module and the category of left modules over its universal enveloping crossed module of algebras. Our approach is particularly interesting since the actor in the category of Leibniz crossed modules does not exist in general, so the technique used in the proof for the Lie case cannot be applied. Finally we move on to the framework of the Loday-Pirashvili category ℒℳ

Universal enveloping crossed module of Leibniz crossed modules and representations

\mathcal {LM}

Universal central extensions of

in order to comprehend this universal enveloping crossed module in terms of the Lie crossed modules case.

\mathfrak{sl}(m, n, A)

The universal enveloping algebra functor UL: Lb → Alg, defined by Loday and Pirashvili [1], is extended to crossed modules. Then we construct an isomorphism between the category of representations of a Leibniz crossed module and the category of left modules over its universal enveloping crossed module of algebras. Note that the procedure followed in the proof for the Lie case cannot be adapted, since the actor in the category of Leibniz crossed modules does not always exist.