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Amir Fernández Ouaridi

Publications and source records attributed to Amir Fernández Ouaridi.

15 recordsLinked to original sources
arXiv2026

Commutative algebras are ideals of evolution algebras

We prove that every finite-dimensional commutative algebra can be embedded as an ideal of a finite-dimensional evolution algebra. We then determine the smallest possible dimension of an evolution algebra admitting such an embedding and study some structural properties preserved by the construction. Over fields of characteristic different from two, we relate this minimal dimension to simultaneous Waring decompositions of the quadratic map $x\mapsto x^2$. In particular, for unital commutative algebras, the smallest possible dimension of an evolution algebra containing the given algebra coincides with the Waring rank of its squaring map. As an application, we determine this minimal dimension for the truncated polynomial algebras $\mathbb{F}[t]/(t^n)$. Finally, we study embeddings for which additional structure is preserved, including automorphisms and multiplicative bases.

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arXiv2026

On simple transposed Poisson algebras

We develop a structure theory for transposed Poisson algebras over fields of characteristic different from two. In particular, we prove that every finite-dimensional transposed Poisson algebra over an algebraically closed field decomposes as the direct sum of a unital ideal and a nilpotent ideal. As a consequence, we obtain restrictions on simple transposed Poisson algebras and use them to classify the simple finite-dimensional transposed Poisson algebras over an algebraically closed field of characteristic $p>3$. Precisely, we show that every such algebra has as underlying Lie algebra a Zassenhaus algebra $\mathcal{W}(1;n)$ and is isomorphic to one of the algebras of the family $\mathcal{W}_n(q)$ arising from a mutation of a natural associative commutative structure on $\mathcal{W}(1;n)$. We then study the corresponding isomorphism problem for the family $\mathcal{W}_n(q)$ and determine the irreducible finite-dimensional representations of these simple transposed Poisson algebras $\mathcal{W}_n(q)$ in the unital case. We conclude with some applications to Jordan superalgebras, weak-Leibniz algebras and quasi-Poisson algebras.

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arXiv2024

Abelian Subalgebras and Ideals of Maximal Dimension in Poisson algebras

This paper studies the abelian subalgebras and ideals of maximal dimension of Poisson algebras $\mathcal{P}$ of dimension $n$. We introduce the invariants $α$ and $β$ for Poisson algebras, which correspond to the dimension of an abelian subalgebra and ideal of maximal dimension, respectively. We prove that these invariants coincide if $α(\mathcal{P}) = n-1$. We characterize the Poisson algebras with $α(\mathcal{P}) = n-2$ over arbitrary fields. In particular, we characterize Lie algebras $L$ with $α(L) = n-2$. We also show that $α(\mathcal{P}) = n-2$ for nilpotent Poisson algebras implies $β(\mathcal{P})=n-2$. Finally, we study these invariants for various distinguished Poisson algebras, providing us with several examples and counterexamples.

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arXiv2024

Degenerations of Poisson-type algebras

The degenerations of Poisson-type algebras are studied in the following varieties in dimension two: Leibniz--Poisson algebras, transposed Leibniz--Poisson algebras, Novikov--Poisson algebras, commutative pre-Lie algebras, anti-pre-Lie Poisson algebras and pre-Poisson algebras. For these varieties, the algebraic and geometric classifications are given. Also, the complete graph of degenerations is obtained, together with the description of the orbit closures of each of its algebras and parametric families up to isomorphism.

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arXiv2023

Zinbiel superalgebras

Throughout the current paper, we extend the study of Zinbiel algebras to Zinbiel superalgebras. In particular, we show that all the Zinbiel superalgebras over an arbitrary field are nilpotent in the same way as occurs for Zinbiel algebras. Moreover and since the most important cases of nilpotent algebras or superalgebras are those with maximal nilpotency index, we study the complex null-filiform Zinbiel superalgebra, i.e. the only one single generated, proving that is unique up to isomorphism. After that, we characterise the naturally graded filiform ones and obtain low-dimensional classifications.

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arXiv2023

On the simple transposed Poisson algebras and Jordan superalgebras

We prove that a transposed Poisson algebra is simple if and only if its associated Lie bracket is simple. Consequently, any simple finite-dimensional transposed Poisson algebra over an algebraically closed field of characteristic zero is trivial. Similar results are obtained for transposed Poisson superalgebras. Furthermore, we show that the Kantor double of a transposed Poisson algebra is a Jordan superalgebra. Additionally, a simplicity criterion for the Kantor double of a transposed Poisson algebra is obtained.

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arXiv2023

Non-degenerate evolution algebras

In this paper we introduce a new invariant for a non-degenerate evolution algebra, which consists of an ordered sequence of evolution algebras of lower dimension, belonging all of them to a specific family. We use this invariant to propose a method to classify non-degenerate evolution algebras, and we apply it up to dimension 3. We also use it to describe the derivations of some families of evolution algebras and the variety of evolution algebras with square not greater than 1.

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arXiv2022

Conservative algebras of $2$-dimensional algebras, IV

The notion of conservative algebras appeared in a paper by Kantor in 1972. Later, he defined the conservative algebra $W(n)$ of all algebras (i.e. bilinear maps) on the $n$-dimensional vector space. If $n>1$, then the algebra $W(n)$ does not belong to any well-known class of algebras (such as associative, Lie, Jordan, or Leibniz algebras). It looks like $W(n)$ in the theory of conservative algebras plays a similar role to the role of $\mathfrak{gl}_n$ in the theory of Lie algebras. Namely, an arbitrary conservative algebra can be obtained from a universal algebra $W(n)$ for some $n \in \mathbb{N}.$ The present paper is a part of a series of papers, dedicated to the study of the algebra $W(2)$ and its principal subalgebras.

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arXiv2022

Degenerations of Poisson algebras

We construct a method to obtain the algebraic classification of Poisson algebras defined on a commutative associative algebra, and we apply it to obtain the classification of the $3$-dimensional Poisson algebras. In addition, we study the geometric classification, the graph of degenerations and the closures of the orbits of the variety of $3$-dimensional Poisson algebras. Finally, we also study the algebraic classification of the Poisson algebras defined on a commutative associative null-filiform or filiform algebra and, to enrich this classification, we study the degenerations between these particular Poisson algebras.

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arXiv2022

The algebraic and geometric classification of nilpotent Lie triple systems up to dimension four

In this paper we generalize the Skjelbred Sund method, used to classify nilpotent Lie algebras, in order to classify triple systems with non zero annihilator. We develop this method with the purpose of classifying nilpotent Lie triple systems, obtaining from it the algebraic classification of the nilpotent Lie triple systems up to dimension four. Additionally, we obtain the geometric classification of the variety of nilpotent Lie triple systems up to dimension four.

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arXiv2019

Degenerations of nilpotent algebras

We give a complete description of degenerations of $3$-dimensional nilpotent algebras, $4$-dimensional nilpotent commutative algebras and $5$-dimensional nilpotent anticommutative algebras over $ \mathbb C$. In particular, we correct several mistakes from the paper `Contractions of low-dimensional nilpotent Jordan algebras' by Ancochea Bermúdez, Fresán and Margalef Bentabol.

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arXiv2018

The classification of $2$-dimensional rigid algebras

Using the algebraic classification of all $2$-dimensional algebras, we give the algebraic classification of all $2$-dimensional rigid, conservative and terminal algebras over an algebraically closed field of characteristic 0. We have the geometric classification of the variety of $2$-dimensional terminal algebras, and based on the geometric classification of these algebras we formulate some open problems.

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arXiv2018

Classification of bilinear maps with radical of codimension 2

Let ${\mathbb V}$ be an $n$-dimensional linear space over an algebraically closed base field. We provide a classification, up to equivalence, of all of the bilinear maps $f:{\mathbb V} \times {\mathbb V} \to {\mathbb V}$ such that $dim(rad(f)) =n-2$. This is equivalent to give a complete classification (up to isomorphism) of all $n$-dimensional algebras with annihilator of dimension $n-2$ or, in other words, a classification of the annihilator extensions of all $2$-dimensional algebras.

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