arXiv subjects

J. M. Sánchez

Publications and source records attributed to J. M. Sánchez.

3 recordsLinked to original sources

arXiv2024

On naturally graded Lie and Leibniz superalgebras

In general, the study of gradations has always represented a cornerstone in algebra theory. In particular, \textit{naturally graded} seems to be the first and the most relevant gradation when it comes to nilpotent algebras, a large class of solvable ones. In fact, many families of relevant solvable algebras have been obtained by extensions of naturally graded nilpotent algebras, i.e. solvable algebras with a well-structured nilradical. Thus, the aim of this work is introducing the concept of naturally graded for superalgebra structures such as Lie and (non-Lie) Leibniz. After having defined naturally graded Lie and Leibniz superalgebras, we characterize natural gradations on a very important class of each of them, that is, those with maximal super-nilindex.

math.RA↗

arXiv2024

Odd-quadratic Lie superalgebras with a weak filiform module as an odd part

The aim of this work is to study a very special family of odd-quadratic Lie superalgebras ${\mathfrak g}={\mathfrak g}_{\bar 0}\oplus {\mathfrak g}_{\bar 1}$ such that ${\mathfrak g}_{\bar 1}$ is a weak filiform ${\mathfrak g}_{\bar 0}$-module (weak filiform type). We introduce this concept after having proved that the unique non-zero odd-quadratic Lie superalgebra $({\mathfrak g},B)$ with ${\mathfrak g}_{\bar 1}$ a filiform ${\mathfrak g}_{\bar 0}$-module is the abelian $2$-dimensional Lie superalgebra ${\mathfrak g}={\mathfrak g}_{\bar 0} \oplus {\mathfrak g}_{\bar 1}$ such that $\mbox{\rm dim }{\mathfrak g}_{\bar 0}=\mbox{\rm dim }{\mathfrak g}_{\bar 1}=1$. Let us note that in this context the role of the center of ${\mathfrak g}$ is crucial. Thus, we obtain an inductive description of odd-quadratic Lie superalgebras of weak filiform type via generalized odd double extensions. Moreover, we obtain the classification, up to isomorphism, for the smallest possible dimensions, that is, six and eight.

math.RT↗

arXiv2024

Leibniz algebras and graphs

We consider a Leibniz algebra ${\mathfrak L} = {\mathfrak I} \oplus {\mathfrak V}$ over an arbitrary base field $\mathbb{F}$, being ${\mathfrak I}$ the ideal generated by the products $[x,x], x \in {\mathfrak L}$. This ideal has a fundamental role in the study presented in our paper. A basis $\B=\{v_i\}_{i \in I}$ of ${\mathfrak L}$ is called multiplicative if for any $i,j \in I$ we have that $[v_i,v_j] \in {\mathbb F}v_k$ for some $k \in I$. We associate an adequate graph $Γ({\mathfrak L},\B)$ to ${\mathfrak L}$ relative to $\B$. By arguing on this graph we show that ${\mathfrak L}$ decomposes as a direct sum of ideals, each one being associated to one connected component of $Γ({\mathfrak L},\B)$. Also the minimality of ${\mathfrak L}$ and the division property of ${\mathfrak L}$ are characterized in terms of the weak symmetry of the defined subgraphs $Γ({\mathfrak L},\B_{\mathfrak I})$ and $Γ({\mathfrak L},\B_{\mathfrak V})$.

math.RT↗