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María Alejandra Alvarez

Publications and source records attributed to María Alejandra Alvarez.

8 recordsLinked to original sources
arXiv2025

Polynomial invariants for low dimensional algebras

We classify all two-dimensional simple algebras (which may be non-associative) over an algebraically closed field. For each two-dimensional algebra $\mathcal{A}$, we describe a minimal (with respect to inclusion) generating set for the algebra of invariants of the $m$-tuples of $\mathcal{A}$ in the case of characteristic zero. In particular, we establish that for any two-dimensional simple algebra $\mathcal{A}$ with a non-trivial automorphism group, the Artin--Procesi--Iltyakov Equality holds for $\mathcal{A}^m$; that is, the algebra of polynomial invariants of $m$-tuples of $\mathcal{A}$ is generated by operator traces. As a consequence, we describe two-dimensional algebras that admit a symmetric or skew-symmetric invariant nondegenerate bilinear form.

math.RA↗
arXiv2022

The algebraic and geometric classification of Zinbiel algebras

This paper is devoted to the complete algebraic and geometric classification of complex $5$-dimensional Zinbiel algebras. In particular, we proved that the variety of complex $5$-dimensional Zinbiel algebras has dimension $24$, it is defined by $16$ irreducible components and it has $11$ rigid algebras.

math.RA↗
arXiv2022

The algebraic and geometric classification of nilpotent weakly associative and symmetric Leibniz algebras

This paper is devoted to the complete algebraic and geometric classification of complex $4$-dimensional nilpotent weakly associative, complex $4$-dimensional symmetric Leibniz algebras, and complex $5$-dimensional nilpotent symmetric Leibniz algebras. In particular, we proved that the variety of complex $4$-dimensional symmetric Leibniz algebras has no Vergne--Grunewald--O'Halloran Property (there is an irreducible component formed by only nilpotent algebras), but on the other hand, it has Vergne Property (there are no rigid nilpotent algebras).

math.RA↗
arXiv2021

Central extensions of 3-dimensional Zinbiel algebras

We describe all central extensions of all $3$-dimensional non-zero complex Zinbiel algebras. As a corollary, we have a full classification of $4$-dimensional non-trivial complex Zinbiel algebras and a full classification of $5$-dimensional non-trivial complex Zinbiel algebras with $2$-dimensional annihilator, which gives the principal step in the algebraic classification of $5$-dimensional Zinbiel algebras.

math.RA↗
arXiv2019

Varieties of Nilpotent Lie Superalgebras of dimension $\leq 5$

In this paper we study the varieties of nilpotent Lie superalgebras of dimension $\leq 5$. We provide the algebraic classification of these superalgebras and obtain the irreducible components in every variety. As a by product we construct rigid nilpotent Lie superalgebras of arbitrary dimension.

math.RA↗
arXiv2018

On Degenerations of Lie Superalgebras

We give necessary conditions for the existence of degenerations between two complex Lie superalgebras of dimension $(m,n)$. As an application, we study the variety $\mathcal{LS}^{(2,2)}$ of complex Lie superalgebras of dimension $(2,2)$. First we give the algebraic classification and then obtain that $\mathcal{LS}^{(2,2)}$ is the union of seven irreducible components, three of which are the Zariski closures of rigid Lie superalgebras. As byproduct, we obtain an example of a nilpotent rigid Lie superalgebra, in contrast to the classical case where no example is known.

math.RA↗
arXiv2017

Degenerations of Jordan Superalgebras

We describe degenerations of three-dimensional Jordan superalgebras over $\mathbb{C}$. In particular, we describe all irreducible components in the corresponding varieties.

math.RA↗