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Sofiane Bouarroudj

Publications and source records attributed to Sofiane Bouarroudj.

At least 19 recordsLinked to original sources
arXiv2026

Structure of Flat Quadratic Quasi-Frobenius Lie Superalgebras via Double Extensions

A flat quadratic quasi-Frobenius Lie superalgebra is a quadratic Lie superalgebra equipped with an additional symplectic structure that is flat with respect to the natural symplectic product. In this paper, we introduce the notion of a flat quadratic double extension of a flat quadratic quasi-Frobenius Lie superalgebra, in the cases where both the symplectic structure and the quadratic structure are either even or odd. We show that, over an algebraically closed field, any such Lie superalgebra can be constructed through a sequence of flat quadratic double extensions starting from the trivial algebra $\{0\}$. Moreover, when the quadratic and symplectic structures have different parity, we introduce the notion of a planar double extension, which constitutes the main novelty of this paper. In this case, we prove that such Lie superalgebras have total dimension $4n$. Finally, we classify flat quadratic quasi-Frobenius Lie superalgebras of dimension at most four and present explicit examples in dimensions six and eight.

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arXiv2026

Pseudo-Euclidean Novikov Superalgebras: Structure and Properties

A pseudo-Euclidean Novikov superalgebra $A$ is a Novikov superalgebra endowed with a non-degenerate symmetric bilinear form $\langle,\rangle$ such that all left multiplication operators are $\langle,\rangle$-antisymmetric. In this case, the associated Lie superalgebra $(A^{-},$\langle,\rangle$)$ is a flat pseudo-Euclidean Lie superalgebra. In this paper, we investigate the structure of pseudo-Euclidean Novikov superalgebras. In particular, we introduce a distinguished subclass, called Milnor superalgebras, and prove that any pseudo-Euclidean Novikov superalgebra whose two-sided ideal is non-degenerate belongs to this class. We provide a method for constructing pseudo-Euclidean Novikov superalgebras. We also introduce a double extension procedure for pseudo-Euclidean Novikov superalgebras and show that every such superalgebra with a degenerate two-sided ideal can be obtained via this method. Furthermore, we establish that any pseudo-Euclidean Novikov superalgebra is either a Milnor superalgebra or can be obtained by a sequence of double extensions starting from a Milnor superalgebra. As an application, we provide a complete classification of pseudo-Euclidean Novikov superalgebras of total dimension at most four.

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arXiv2026

Hyper relative differential operators on Lie algebras

In this paper, we first introduce the notion of a hyper relative differential operator on a Lie algebra, in which Nijenhuis operators are used to characterize the relative differential operators and their inverse. We then introduce the notions of DN-structures, KN-structures, and KD-structures on Lie algebras and study the relationships between DN-structures, KD-structures, KN-structures, and hyper relative differential operators. Finally, we investigate hyper symplectic structures and hyper Hessian structures from the view point of hyper relative differential operators, and provide equivalent descriptions for both hyper symplectic structures and hyper Hessian structures.

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arXiv2026

Computer-aided study of double extensions of restricted Lie superalgebras preserving the non-degenerate closed 2-forms in characteristic 2

A Lie (super)algebra with a non-degenerate invariant symmetric bilinear form $B$ is called a nis-(super)algebra. The double extension $\mathfrak{g}$ of a nis-(super)algebra $\mathfrak{a}$ is the result of simultaneous adding to $\mathfrak{a}$ a central element and a derivation so that $\mathfrak{g}$ is a nis-algebra. Loop algebras with values in simple complex Lie algebras are most known among the Lie (super)algebras suitable to be doubly extended. In characteristic 2 the notion of double extension acquires specific features. Restricted Lie (super)algebras are among the most interesting modular Lie superalgebras. In characteristic 2, using Grozman's Mathematica-based package SuperLie, we list double extensions of restricted Lie superalgebras preserving the non-degenerate closed 2-forms with constant coefficients. The results are proved for the number of indeterminates ranging from 4 to 7 - sufficient to conjecture the pattern for larger numbers. Considering multigradings allowed us to accelerate computations up to 100 times.

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arXiv2026

Flat quasi-Frobenius Lie superalgebras

A non-associative superalgebra is called pre-symplectic if it is equipped with a non-degenerate, anti-symmetric bilinear form. It is called quasi-Frobenius if, in addition, is a Lie superalgebra and the form is closed. We introduce the Levi-Civita product associated with pre-symplectic superalgebras and establish its existence and uniqueness. We then introduce the symplectic product associated with quasi-Frobenius Lie superalgebras. We prove that while such a product always exists, it is not unique. We therefore define a natural symplectic product that depends only on the Lie structure and the bilinear form. When the curvature of this product vanishes, the superalgebra is called a flat quasi-Frobenius Lie superalgebra. In this paper, we study flat quasi-Frobenius Lie superalgebras and introduce the notion of a flat double extension. We prove that the double extension process characterizes such superalgebras. More precisely, every flat orthosymplectic (resp. periplectic) quasi-Frobenius Lie superalgebra can be obtained by a sequence of flat double extensions starting from an abelian one (resp. the trivial one). Moreover, we show that every flat quasi-Frobenius Lie superalgebra is nilpotent with a degenerate center. We apply our results to obtain a complete classification of such superalgebras of total dimension at most five.

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arXiv2025

The Superization of Hochschild's Lemma and Restricted Lie-Rinehart Superalgebras

The main goal of this paper is to introduce the notion of restricted Lie-Rinehart superalgebra over a field of characteristic $p>2$, motivated by a generalization of Hochschild's lemma to the super setting. We extend Schauenburg's proof of Hochschild's lemma to Lie-Rinehart superalgebras and we prove a superized version that serves as a foundation for our construction. Building upon this, we define restricted Lie-Rinehart superalgebras, investigate their representations, construct the semi-direct product with a restricted module, and provide several examples. Finally, we construct the corresponding universal enveloping algebra and show that this algebra satisfies the expected universal property.

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arXiv2025

Flat pseudo-Euclidean Leibniz superalgebras

In this paper, we introduce pre-Lie and pre-Leibniz superalgebras, which generalize pre-Lie and pre-Leibniz algebras to the super setting. Additionally, we define a Levi-Civita product associated with a symmetric non-degenerate bilinear form on a non-associative superalgebra. This leads to the definition of flat pseudo-Euclidean left Leibniz superalgebras as those whose Levi-Civita product induces a pre-Leibniz structure. We study the structure of flat pseudo-Euclidean left Leibniz superalgebras and provide a characterization theorem. In the second part, we focus on quadratic Leibniz superalgebras and show that such a superalgebra is flat if and only if it is symmetric Leibniz and 2-step nilpotent. We further study the structure of quadratic 2-step nilpotent symmetric Leibniz superalgebras. Finally, we introduce the notion of double extension for flat pseudo-Euclidean (resp. Lie) left Leibniz superalgebras and prove that any flat pseudo-Euclidean non-Lie left Leibniz superalgebra can be obtained by a sequence of double extensions starting from a flat pseudo-Euclidean Lie superalgebra.

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arXiv2025

On Zinbiel and Tortkara superalgebras

In this paper, we study Zinbiel superalgebras and special Tortkara superalgebras, highlighting key differences between the super and the non-super setting. We present examples of Zinbiel superalgebras with Rota-Baxter operators and construct a basis for free Zinbiel superalgebras. Moreover, we establish a superalgebraic analogue of the Lie criterion for Zinbiel superalgebras. In contrast to the classical case, some homomorphic images of special Tortkara superalgebras on two generators are exceptional. Finally, we present a classification of all Tortkara superalgebras of dimensions 2 and 3.

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arXiv2025

Cohomology of Restricted Poisson algebras in characteristic 2

In this paper, we study restricted Poisson algebras in characteristic 2 and their relationship with restricted Lie-Rinehart algebras, for which we develop a cohomology theory and investigate abelian extensions. We also construct a full cohomology complex for restricted Poisson algebras in characteristic 2 that captures formal deformations and prove that it is isomorphic to the cohomology complex of a suitable restricted Lie-Rinehart algebra, under certain assumptions. A number of examples are provided in order to illustrate our constructions.

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arXiv2025

Left-symmetric superalgebras and Lagrangian extensions of Lie superalgebras in characteristic 2

The purpose of this paper is twofold. First, we introduce the notions of left-symmetric and left alternative structures on superspaces in characteristic 2. We describe their main properties and classify them in dimension 2. We show that left-symmetric structures can be queerified if and only if they are left-alternative. Secondly, we present a method of Lagrangian extension of Lie superalgebras in characteristic 2 with a flat torsion-free connection. We show that any strongly polarized quasi-Frobenius Lie superalgebra can be obtained as a Lagrangian extension. Further, we demonstrate that Lagrangian extensions are classified by a certain cohomology space that we introduce. To illustrate our constructions, all Lagrangian extensions in dimension 4 have been described.

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arXiv2024

Central extensions of restricted Lie superalgebras and classification of $p$-nilpotent Lie superalgebras in dimension $4$

The first main result of this paper is to build the first and second restricted cohomology groups for restricted Lie superalgebras in characteristic $p\geq3$, modifying a construction by Yuan, Chen and Cao. We will explain how these groups capture some algebraic structures, such as extensions and derivations. Further, we apply this construction to classify $p$-nilpotent restricted Lie superalgebras up to dimension $4$ over an algebraically closed field of characteristic $p\geq3$.

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arXiv2024

Double extensions of quasi-Frobenius Lie superalgebras with degenerate center

We develop the process of symplectic double extensions for Lie superalgebras with degenerate center. The construction is a superization of a recent work by Fischer, and generalize our previous work. We provide a standard model for such double extensions, where the symplectic form is either orthosymplectic or periplectic. Additionally, we show that every double extension is naturally equivalent to either of these two standard types of extensions. Several examples in low dimensions are given to illustrate the concept.

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arXiv2023

Symplectic Double Extensions for Restricted Quasi-Frobenius Lie (Super)Algebras

In this paper, we present a method of symplectic double extensions for restricted quasi-Frobenius Lie superalgebras. Certain cocycles in the restricted cohomology represent obstructions to symplectic double extension, which we fully describe. We found a necessary condition for which a restricted quasi-Frobenius Lie superalgebras is a symplectic double extension of a smaller restricted Lie superalgebra. The constructions are illustrated with a few examples.

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arXiv2023

Deformations of Symmetric Simple Modular Lie (Super)Algebras

We say that a Lie (super)algebra is ''symmetric'' if with every root (with respect to the maximal torus) it has the opposite root of the same multiplicity. Over algebraically closed fields of positive characteristics (up to 7 or 11, enough to formulate a general conjecture), we computed the cohomology corresponding to the infinitesimal deformations of all known simple finite-dimensional symmetric Lie (super)algebras of rank $<9$, except for superizations of the Lie algebras with ADE root systems, and queerified Lie algebras, considered only partly. The moduli of deformations of any Lie superalgebra constitute a supervariety. Any infinitesimal deformation given by any odd cocycleis integrable. All deformations corresponding to odd cocycles are new. Among new results are classifications of the cocycles describing deforms (results of deformations) of the 29-dimensional Brown algebra in characteristic 3, of Weisfeiler-Kac algebras and orthogonal Lie algebras without Cartan matrix in characteristic 2. Open problems: describe non-isomorphic deforms and equivalence classes of cohomology theories. Appendix: For several modular analogs of complex simple Lie algebras, and simple Lie algebras indigenous to characteristics 3 and 2, we describe the space of cohomology with trivial coefficients. We show that the natural multiplication in this space is very complicated.

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arXiv2023

Derivations and Central Extensions of Symmetric Modular Lie Algebras and Superalgebras

Over algebraically closed fields of positive characteristic, for simple Lie (super)algebras, and certain Lie (super)algebras close to simple ones, with symmetric root systems (such that for each root, there is minus it of the same multiplicity) and of ranks less than or equal to 8 - most needed in an approach to the classification of simple vectorial Lie superalgebras (i.e., Lie superalgebras realized by means of vector fields on a supermanifold), - we list the outer derivations and nontrivial central extensions. When the conjectural answer is clear for the infinite series, it is given for any rank. We also list the outer derivations and nontrivial central extensions of one series of non-symmetric (except when considered in characteristic 2), namely periplectic, Lie superalgebras - the one that preserves the nondegenerate symmetric odd bilinear form, and of the Lie algebras obtained from them by desuperization. We also list the outer derivations and nontrivial central extensions of an analog of the rank 2 exceptional Lie algebra discovered by Shen Guangyu. Several results indigenous to positive characteristic are of particular interest being unlike known theorems for characteristic 0, some results are, moreover, counterintuitive.

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arXiv2022

Hom-Lie superalgebras in characteristic 2

The main goal of this paper is to develop the structure theory of Hom-Lie superalgebras in characteristic 2. We discuss their representations, semidirect product, $α^k$-derivations and provide a classification in low dimension. We introduce another notion of restrictedness on Hom-Lie algebras in characteristic 2, different from one given by Guan and Chen. This definition is inspired by the process of queerification of restricted Lie algebras in characteristic 2. We also show that any restricted Hom-Lie algebra in characteristic 2 can be queerified to give rise to a Hom-Lie superalgebra. Moreover, we develop a cohomology theory of Hom-Lie superalgebras in characteristic 2, which provides a cohomology of ordinary Lie superalgebras. Furthermore, we establish a deformation theory of Hom-Lie superalgebras in characteristic 2 based on this cohomology.

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arXiv2022

Manin triples and non-degenerate anti-symmetric bilinear forms on Lie superalgebras in characteristic $2$

In this paper, we introduce and develop the notion of a Manin triple for a Lie superalgebra $\mathfrak g$ defined over a field of characteristic $p=2$. We find cohomological necessary conditions for the pair $(\mathfrak g, \mathfrak g^*)$ to form a Manin triple. We introduce the concept of Lie bi-superalgebras for $p=2$ and establish a link between Manin triples and Lie bi-superalgebras. In particular, we study Manin triples defined by a classical $r$-matrix with an extra condition (called an admissible classical $r$-matrix). A particular case is examined where $\mathfrak g$ has an even invariant non-degenerate bilinear form. In this case, admissible $r$-matrices can be obtained inductively through the process of double extensions. In addition, we introduce the notion of double extensions of Manin triples, and show how to get a new Manin triple from an existing one.

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