arXiv subjects

Sania Asif

Publications and source records attributed to Sania Asif.

17 recordsLinked to original sources

arXiv2026

Higher Structures of Rota--Baxter Lie $H$-Pseudoalgebras

This paper investigates Rota--Baxter Lie $H$-pseudoalgebras. We develop a cohomology theory for $\lambda$-weighted relative Rota--Baxter operators via a Maurer--Cartan approach, constructing the underlying differential graded Lie algebra. We classify non-abelian extensions using second cohomology and derive the Wells exact sequence to address the inducibility of automorphisms. Furthermore, we explore the homotopy theory of these structures by introducing $2$-term skeletal and strict Rota--Baxter $L_\infty$-$H$-pseudoalgebras. In particular, we establish a one-to-one correspondence between strict $2$-term structures and crossed modules of Rota--Baxter Lie $H$-pseudoalgebras. These results establish a foundational framework for future advancements in the higher categorical theory of pseudoalgebras with algebraic operators. Ultimately, this work provides a robust foundation for the higher categorical study of pseudoalgebras equipped with algebraic operators.

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arXiv2025

Quasi-Twilled Lie Pseudolgebras and Their Deformation Maps

In this paper, we present a unified framework for studying cohomology theories of various operators in the context of pseudoalgebras. The central tool in our approach is the notion of a quasi-twilled Lie pseudoalgebra. We introduce two types of deformation maps. Type I unifies modified $r$ matrices, crossed homomorphisms, derivations, and homomorphisms; and Type II provides a uniform treatment of relative Rota-Baxter operators, twisted Rota-Baxter operators, Reynolds operators, and deformation maps of matched pairs of Lie conformal algebras. We construct the corresponding controlling algebras and define cohomology theories for both types of deformation maps. These results recover existing cohomological results for known operators and yield new results, including the cohomology theory for modified $r$-matrices and deformation maps of matched pairs of Lie pseudoalgebras.

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arXiv2025

Isoclinism in regular Hom-Lie Yamaguti algebras

In this paper, we develop the theory of \emph{isoclinism} for regular Hom-Lie Yamaguti algebras, a class that unifies several generalizations of Lie algebras. Although isomorphism implies isoclinism by definition, the converse is not true in general. We introduce the notion of a \emph{factor set} and use it to analyze the structure of isoclinism families. Our main result establishes that for finite-dimensional regular Hom-Lie Yamaguti algebras of the same dimension, isoclinism implies isomorphism. This generalizes recent classification theorems for Lie-Yamaguti algebras and Hom-Lie superalgebras, highlighting a strong rigidity property in the finite-dimensional setting. The proof relies on the existence of stem algebras and a decomposition theorem within isoclin families.

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arXiv2025

Cohomology, Homotopy, Extensions, and Automorphisms of Nijenhuis Lie Conformal Algebras

This paper explores various algebraic and homotopical aspects of Nijenhuis Lie conformal algebras, including their cohomology theory, $\mathcal{L}_\infty$-structures, non-abelian extensions, and automorphism groups. We define the cohomology of a Nijenhuis Lie conformal algebra and relate it to the deformation theory of such structures. We also introduce $2$-term Nijenhuis $\mathcal{L}_\infty$-conformal algebras and establish their correspondence with crossed modules and $3$-cocycles in the cohomology of Nijenhuis Lie conformal algebras. Furthermore, we develop a classification theory for non-abelian extensions of Nijenhuis Lie conformal algebras via the second non-abelian cohomology group. Finally, we study the inducibility problem for automorphisms under such extensions, introducing a Wells-type map and deriving an associated exact sequence.

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arXiv2024

Exploring Cohomology, Deformations, and Hom-NS Structures in Hom-Leibniz Conformal Algebras through Nijenhuis Operators

This paper studies the Nijenhuis operator on Hom-Leibniz conformal algebra, defining their representations and cohomologies. We determine the cohomologies for both Hom-Leibniz conformal algebra and Nijenhuis operators on Hom-Leibniz conformal algebra. Subsequently, establishing the cohomology of Hom-Nijenhuis-Leibniz conformal algebras. As an application to this cohomology, we study formal deformations of the Nijenhuis operator on Hom-Leibniz conformal algebra. Additionally, we introduce Hom-NS-Leibniz conformal algebra and explore how various operators such as Rota-Baxter operator, Twisted Rota Baxter operator, and Nijenhuis operators can provide Hom-NS-Leibniz conformal algebras.

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arXiv2024

Cohomology and Homotopification of averaging operators on the Lie conformal algebras

Building upon the work of Pavel in [P. Kolesnikov, Journal of Mathematical Physics, 56, 7 (2015)], we first present the cohomology of averaging operators on the Lie conformal algebras and use it to develop the cohomology of averaging Lie conformal algebras. We then introduce the homotopy version of averaging Lie conformal algebras and establish a connection between $2$-term averaging $\mathfrak{L}_\infty$-conformal algebra with the $3$-cocycle and crossed module of averaging Lie conformal algebra. Next, we study the non-abelian extension of the averaging Lie conformal algebras, showing that they are classified by the second non-abelian cohomology group. Finally, we demonstrate that a pair of automorphisms of averaging Lie conformal algebra is inducible if it can be seen as an image of a suitable Wells map.

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arXiv2023

Cohomology and deformation theory of $\mathcal{O}$-operators on Hom-Lie conformal algebras

In the present paper, we aim to introduce the cohomology of $\mathcal{O}$-operators defined on the Hom-Lie conformal algebra concerning the given representation. To obtain the desired results, we describe three different cochain complexes and discuss the interrelation of their coboundary operators. And show that differential maps on the graded Lie algebra can also be defined by using the Maurer-Cartan element. We further find out that, the $\mathcal{O}$-operator on the given Hom-Lie conformal algebra serves as a Maurer-Cartan element and it leads to acquiring the notion of a differential map in terms of $\mathcal{O}$-operator $\delta_{\mathcal{T}}$. Next, we provide the notion of Hom-pre-Lie conformal algebra, that induces a sub-adjacent Hom-Lie conformal algebra structure. The differential $\delta_{\beta,\alpha}$ of this sub-adjacent Hom-Lie conformal algebra is related to the differential $\delta_{\mathcal{T}}$. Finally, we provide the deformation theory of $\mathcal{O}$-operators on the Hom-Lie conformal algebras as an application to the cohomology theory, where we discuss linear and formal deformations in detail.

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arXiv2023

Rota-Baxter operators and Loday-type algebras on the BiHom-associative conformal algebras

(Tri)dendriform algebras, Rota-Baxter operators, and closely related NS-algebras have a number of dominant applications in physics, especially in quantum field theory. Proceeding from the recent study relating these structures, this paper considers (tri)dendriform algebras, NS-algebras, and (twisted)Rota-Baxter operators in the context of BiHom-associative conformal algebras. A comprehensive investigation of the BiHom-(tri)dendriform conformal algebras and their characterization in terms of conformal bimodule has been conducted. The study of BiHom-NS-conformal algebra reveals that it is not only a generalization of NS-conformal algebra using two structural maps but is also the generalization of BiHom-(tri)dendriform conformal algebras. Additionally, it is found to have a close proximity between BiHom-twisted Rota-Baxter operators and BiHom-NS-conformal algebras. The comparative study to Rota-Baxter operators on BiHom-associative conformal algebras and Rota-Baxter operators on BiHom-(tri)dendriform conformal algebras reveals a relationship between BiHom-quadri conformal algebra and Rota-Baxter operators. In the end, the concept of Rota-Baxter system (a generalization of the Rota-Baxter operator) for BiHom-associative conformal algebras and BiHom-dendriform conformal algebras is narrated, where the interconnections of these algebras are depicted. Furthermore, a connection is established between BiHom-quadri conformal algebras and Rota-Baxter systems for BiHom-dendriform conformal algebras.

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arXiv2023

BiHom-(pre-)Poisson conformal algebra

The aim of this study is to introduce the notion of BiHom-Poisson conformal algebra, BiHom-pre-Poisson conformal algebra, and their related structures. We show that we can construct many new BiHom-Poisson conformal algebras for a given BiHom-Poisson conformal algebra. Moreover, the tensor product of two BiHom-Poisson conformal algebras is also a BiHom-Poisson conformal algebra. We further describe the conformal bimodule and representation theory of BiHom-Poisson conformal algebra. In addition, we define BiHom-pre-Poisson conformal algebra as the combination of BiHom-preLie conformal algebra and BiHom-dendriform conformal algebra under some compatibility conditions. We also demonstrate that how to construct BiHom-Poisson conformal algebra from BiHom-pre-Poisson conformal algebra and provide the representation theory for BiHom-pre-Poisson conformal algebra. Finally, a detailed description of $\mathcal{O}$-operators and Rota-Baxter operators on BiHom-Poisson conformal algebra is provided.

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arXiv2023

Cohomology of Twisted Rota-Baxter operators on Associative~Conformal Algebra

In this paper, we examine the concept of twisted Rota-Baxter (TRB) operators on associative conformal algebras. Our strategy begins by constructing an $L_\infty$-algebra using Maurer-Cartan elements derived from $H$-twisted Rota-Baxter ($H$-TRB) operators on associative conformal algebras. This structure leads us to explore the cohomology of the conformal $H$-TRB operator, which is characterized as the Hochschild cohomology of a specific associative conformal algebra with coefficients in a conformal bimodule. Furthermore, we study the linear and formal deformations of conformal $H$-TRB operators to explore the application of cohomology.

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arXiv2023

On the Conformal biderivations and conformal commuting maps on the current Lie Conformal superalgebras

Let $L$ be a Lie conformal superalgebra and $A$ be an associative commutative algebra with unity. We define the current Lie conformal superalgebra by the tensor product $L \otimes A.$ We prove every conformal super-biderivation $\varphi_{\lambda }$ on $L$ is of the form of the centroid $Cent(L)$. Moreover, we show that every Lie conformal super-biderivation on $L\otimes A$ also has the same performance as $L$. We also prove that every Lie conformal linear super-commuting map $\varPsi_{\lambda }$ on $L \otimes A$ belongs to $Cent(L \otimes A)$, if the same holds for $L$ as well.

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arXiv2023

Derivations, Cohomology and Deformation of BiHom-Associative Dialgebra

Due to the immense importance of BiHom Type algebras and cohomology of various algebraic structures, this paper is devoted to defining the BiHom-associative dialgebra, its derivation, generalized derivation, and quasi-derivation. We provided the complete classification of these derivations of $2-$ and $3$-dimensional BiHom-associative dialgebras. We further generalized the cohomology of BiHom-associative algebras to the cohomology of BiHom-associative dialgebras. As an application to cohomology, we evaluate the one-parameter formal deformation of BiHom-associative dialgebras.

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arXiv2023

Classification of tridendriform algebra and related structures

The classification of algebraic structures and their derivations is an important and ongoing research area in mathematics and physics, and various results have been obtained in this field. This article presents the classification of tridendriform algebras that was first studied by Loday and Ronco, including an analysis of structure constant equations using computer algebra software. We further explicitly classify the derivations and centroids of tridendriform algebras, showing that there are only trivial derivations for $2$- and $3$-dimensional algebras but $21$ non-isomorphic derivations for $4$-dimensional tridendriform algebras with dimension range from $1$ to $5$. Additionally, for centroids (centroid and quasi-centroid), there are trivial isomorphism classes for $2$ dimensional tridendriform algebra, $6$ non-isomorphic classes for $3$-dimensional tridendriform algebras and $21$ for $4$-dimensional algebras. The dimensions range for centroid is from $1$ to $5$, whereas it is from $1$ to $10$ for quasi-centroid.

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arXiv2023

On some derivations of Lie conformal superalgebras

Let $\mathcal{R}$ be a Lie conformal superalgebra. In this paper, we first investigate the conformal derivation algebra $CDer(\mathcal{R})$, the conformal triple derivation algebra $CTDer(\mathcal{R})$, and the generalized conformal triple derivation algebra $GCTDer(\mathcal{R})$. Moreover, we determine the connection of these derivation algebras. Next, we give a complete classification of the (generalized) conformal triple derivation algebra on all finite simple Lie conformal superalgebras. More specifically, $CTDer(\mathcal{R})=CDer(\mathcal{R})$, where $\mathcal{R}$ is a finite simple Lie conformal superalgebra, but for $GCTDer(\mathcal{R})$, we obtain a conclusion that is closely related to $CDer(\mathcal{R})$. Furthermore, we evaluate the $(\varPhi, \varPsi)$-Lie triple derivations on Lie conformal superalgebra, where $\varPhi$ and $\varPsi$ are associated automorphism of $\phi_{x}\in gc(\mathcal R)$. We evaluated some fundamental properties of $(\varPhi, \varPsi)$- Lie triple derivations. Later, we introduce the definition of $(A, B, C, D)$-derivation on Lie conformal superalgebra. We obtain the relationships between the generalized conformal triple derivations and the conformal $(A, B, C, D)$-derivations on Lie conformal superalgebra. Finally, we have presented the triple homomorphism of Lie conformal superalgebras.

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arXiv2023

Conformal triple derivations and triple homomorphisms of Lie conformal algebras

Let $\mathcal{R}$ be a finite Lie conformal algebra. In this paper, we first investigate the conformal derivation algebra $CDer(\mathcal{R})$, the conformal triple derivation algebra $CTDer(\mathcal{R})$ and the generalized conformal triple derivation algebra $GCTDer(\mathcal{R})$. Mainly, we focus on the connections among these derivation algebras. Next, we give a complete classification of (generalized) conformal triple derivation algebras on all finite simple Lie conformal algebras. In particular, $CTDer(\mathcal{R})= CDer(\mathcal{R})$, where $\mathcal{R}$ is a finite simple Lie conformal algebra. But for $GCDer(\mathcal{R})$, we obtain a conclusion that is closely related to $CDer(\mathcal{R})$. Finally, we introduce the definition of triple homomorphism of a Lie conformal algebra. Furthermore, triple homomorphisms of all finite simple Lie conformal algebras are also characterized.

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arXiv2022

On generalized derivations of polynomial vector fields Lie algebras

In this paper, we study the generalized derivation of a Lie sub-algebra of the Lie algebra of polynomial vector fields on $\mathbb{R}^n$ where $n\geq1$, containing all constant vector fields and the Euler vector field, under some conditions on this Lie sub-algebra.

math.DG↗

arXiv2022

On the cohomology based on the generalized representations of $n$-Lie Algebras

In the present paper, we define the new class of representation on $n$-Lie algebra that is called as generalized representation. We study the cohomology theory corresponding to generalized representations of $n$-Lie algebras and show its relation with the cohomology corresponding to the usual representations. Furthermore, we provide the computation for the low dimensional cocycles.

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