arXiv subjects

Hani Abdelwahab

Publications and source records attributed to Hani Abdelwahab.

At least 19 recordsLinked to original sources

arXiv2026

$δ$-Poisson and transposed $δ$-Poisson algebras

We present a comprehensive study of two new Poisson-type algebras. Namely, we are working with $δ$-Poisson and transposed $δ$-Poisson algebras. Our research shows that these algebras are related to many interesting identities. In particular, they are related to shift associative algebras, $F$-manifold algebras, algebras of Jordan brackets, etc. We classify simple $δ$-Poisson and transposed $δ$-Poisson algebras and found their depolarizations. We study $δ$-Poisson and mixed-Poisson algebras to be Koszul and self-dual. Bases of the free $δ$-Poisson and mixed-Poisson algebras generated by a countable set $X$ are constructed.

math.RA↗

arXiv2026

The algebraic and geometric classification of $δ$-Novikov algebras

The notion of $δ$-Novikov algebras was introduced recently as a generalization of Novikov and bicommutative algebras. It looks like $δ$-Novikov algebras have a richer structure than Novikov algebras. So, unlike Novikov algebras, they have a $2$-dimensional simple algebra for $δ=-1.$ The present paper is dedicated to the study of $3$-dimensional $δ$-Novikov algebras for $δ\notin \big\{0,1\big\}.$ The algebraic and geometric classifications of complex $3$-dimensional $δ$-Novikov algebras are given. As a corollary, we prove that there are no simple $3$-dimensional $δ$-Novikov algebras.

math.RA↗

arXiv2026

The algebraic and geometric classification of noncommutative Jordan superalgebras

The algebraic and geometric classifications of complex $3$-dimensional noncommutative Jordan superalgebras are given. In particular, we obtain the algebraic and geometric classification of $3$-dimensional Kokoris and standard superalgebras, and, due to one-to-one correspondences between suitable superalgebras, we have classifications for generic Poisson-Jordan and generic Poisson superalgebras. As a byproduct, we have the algebraic and geometric classification of the variety of $3$-dimensional anticommutative superalgebras and its principal subvarieties: Lie, Malcev, binary Lie, Tortkara, anticommutative $\mathfrak{CD}$-, $\mathfrak{s}_4$-, anticommutative terminal superalgebras, anticommutative conservative and anticommutative quasi-conservative $\big($rigid$\big)$ superalgebras.

math.RA↗

arXiv2026

The algebraic and geometric classification of right alternative superalgebras

The algebraic and geometric classifications of complex $3$-dimensional right alternative superalgebras are given. As a byproduct, we have the algebraic and geometric classification of the variety of $3$-dimensional $\mathfrak{perm}$, binary $\mathfrak{perm}$, associative, binary associative, $\big(-1,1\big)$-, and binary $\big(-1,1\big)$-superalgebras.

math.RA↗

arXiv2026

The algebraic and geometric classification of commutative post-Lie algebras

We study commutative post-Lie algebras $(${\rm CPA}s$)$ from an algebraic point of view. Firstly, we find some new identities in {\rm CPA}, which shows that the commutative multiplication gives a medial and derived commutative associative algebra. As corollaries, we have that there are no simple nontrivial commutative post-Lie algebras and that perfect Lie and centrless perfect commutative associative algebras do not admit nontrivial {\rm CPA} structures. The identities of depolarized {\rm CPA}s are defined. Based on the obtained identities, we developed a method for the classification of $n$-dimensional {\rm CPA}s and gave the algebraic classification of $3$-dimensional {\rm CPA}. We also developed another method for classifying $n$-dimensional nilpotent {\rm CPA}s from nilpotent {\rm CPA}s of smaller dimension and gave the algebraic classification of $4$-dimensional nilpotent {\rm CPA}s. Based on the obtained results, we present the geometric classifications of complex $3$-dimensional and $4$-dimensional nilpotent {\rm CPA}s.

math.RA↗

arXiv2026

The algebraic and geometric classification of derived Jordan and bicommutative algebras

We developed a new proper method for classifying $n$-dimensional derived Jordan algebras, and apply it to the classification of $3$-dimensional derived Jordan algebras. As a byproduct, we have the algebraic classification of $3$-dimensional metabelian commutative algebras and $3$-dimensional derived commutative associative algebras. After that, we introduced a method of classifying $n$-dimensional bicommutative algebras, based on the classification of $n$-dimensional derived commutative associative algebras, and applied it to the classification of $3$-dimensional bicommutative algebras. The second part of the paper is dedicated to the geometric classification of $3$-dimensional metabelian commutative, derived commutative associative, derived Jordan and bicommutative algebras.

math.RA↗

arXiv2025

The algebraic and geometric classification of right alternative and semi-alternative algebras

The algebraic and geometric classifications of complex $3$-dimensional right alternative and semi-alternative algebras are given. As corollaries, we have the algebraic and geometric classification of complex $3$-dimensional $\mathfrak{perm}$, binary $\mathfrak{perm}$, associative, $(-1,1)$-, binary $(-1,1)$-, and assosymmetric algebras. In particular, we proved that the first example of non-associative right alternative algebras appears in dimension $3;$ the first example of non-associative assosymmetric algebras appears in dimension $3;$ the first example of non-assosymmetric semi-alternative algebras appears in dimension $4;$ the first example of binary $(-1,1)$-algebras, which is non-$(-1,1)$-, appears in dimension $4;$ the first example of right alternative algebras, which is not binary $(-1,1)$-, appears in dimension $4;$ the first example of binary $\mathfrak{perm}$ non-$\mathfrak{perm}$ algebras appears in dimension $4.$ As a byproduct, we give a more easy answer to problem 2.109 from the Dniester Notebook, previously resolved by Shestakov and Arenas.

math.RA↗

arXiv2025

Classification of $4$-dimensional complex Poisson algebras

The present paper is devoted to the complete classification of $4$-dimensional complex Poisson algebras, taking into account a classification, up to isomorphism, of the complex commutative associative algebras of dimension $4$, as well as by using a Lie algebra classification, up to isomorphism.

math.RT↗

arXiv2025

The algebraic and geometric classification of noncommutative Jordan algebras

In this paper, we develop a method to obtain the algebraic classification of noncommutative Jordan algebras from the classification of Jordan algebras of the same dimension. We use this method to obtain the algebraic classification of complex $3$-dimensional noncommutative Jordan algebras. As a byproduct, we obtain the classification of complex $3$-dimensional Kokoris, standard, generic Poisson, and generic Poisson--Jordan algebras; and also complex $4$-dimensional nilpotent Kokoris and standard algebras. In addition, we consider the geometric classification of varieties of cited algebras, that is the description of its irreducible components.

math.RA↗

arXiv2024

The Algebraic and Geometric Classification of Compatible Pre-Lie Algebras

In this paper, we develop a method to obtain the algebraic classification of compatible pre-Lie algebras from the classification of pre-Lie algebras of the same dimension. We use this method to obtain the algebraic classification of complex 2-dimensional compatible pre-Lie algebras. As a byproduct, we obtain the classification of complex 2-dimensional compatible commutative associative, compatible associative and compatible Novikov algebras. In addition, we consider the geometric classification of varieties of cited algebras, that is the description of its irreducible components.

math.RA↗

arXiv2024

Shift associative algebras

We present a comprehensive study of algebras satisfying the identity $(xy)z=y(zx),$ named as shift associative algebras. Our research shows that these algebras are related to many interesting identities. In particular, they are related to anti-Poisson-Jordan algebras and algebras of associative type $σ$. We study algebras of associative type $σ$ to be Koszul and self-dual. A basis of the free shift associative algebra generated by a countable set $X$ was constructed. An analog of Wedderburn-Artin's theorem was established. The algebraic and geometric classifications of complex $4$-dimensional shift associative algebras are given. In particular, we proved that the first non-associative shift associative algebra appears only in dimension $5$.

math.RA↗

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.

math.RA↗

arXiv2024

Decompositions of linear operators on pre-euclidean spaces by means of graphs

In this work we study a linear operator $f$ on a pre-euclidean space $\mathcal{V}$ by using properties of a corresponding graph. Given a basis $\B$ of $\mathcal{V}$, we present a decomposition of $\mathcal{V}$ as an orthogonal direct sum of certain linear subspaces $\{U_i\}_{i \in I}$, each one admitting a basis inherited from $\B$, in such way that $f = \sum_{i \in I}f_i$, being each $f_i$ a linear operator satisfying certain conditions respect with $U_i$. Considering new hypothesis, we assure the existence of an isomorphism between the graphs associated to $f$ relative to two different bases. We also study the minimality of $\mathcal{V}$ by using the graph associated to $f$ relative to $\B$.

math.RT↗

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.

math.RA↗

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.

math.RA↗

arXiv2019

The algebraic and geometric classification of nilpotent binary Lie algebras

The paper is devoted to give the complete algebraic classification of nilpotent binary Lie algebras of dimension $\leq 6$ over an arbitrary base field ${\mathbb{F}}$ of characteristic not $2$ and the complete geometric classification of nilpotent binary Lie algebras of dimension $6$ over $\mathbb C.$ As an application, we have the algebraic and geometric classification of nilpotent anticommutative $\mathfrak{CD}$-algebras of dimension $\leq 6.$

math.RA↗

arXiv2016

Classification of five-dimensional nilpotent Jordan algebras

The paper is devoted to classify nilpotent Jordan algebras of dimension up to five over an algebraically closed field of characteristic not 2. We obtained a list of 35 isolated non-isomorphic 5-dimensional nilpotent non-associative Jordan algebras and 6 families of non-isomorphic 5-dimensional nilpotent non-associative Jordan algebras depending either on one or two parameters over an algebraically closed field of characteristic not 2 or 3. In addition to these algebras we obtained two non-isomorphic 5-dimensional nilpotent non-associative Jordan algebras over an algebraically closed field of characteristic 3.

math.RA↗

arXiv2015

Five-dimensional nilpotent evolution algebras

The paper is devoted to give a complete classification of five-dimension nilpotent evolution algebras over an algebraically closed field. We obtained a list of 27 isolated non-isomorphic nilpotent evolution algebras and 2 families of non-isomorphic algebras depending on one parameter.

math.AC↗