arXiv subjects

Ripan Saha

Publications and source records attributed to Ripan Saha.

At least 19 recordsLinked to original sources

arXiv2026

Affine Nilpotency and Engel's Theorem for Lie Affgebras

We study some structural properties of Lie affgebras as affine analogues of Lie algebras. We introduce the notion of an ideaf, the affine counterpart of an ideal, and establish characterizations of left and right ideafs. We further study centers, quotient structures, and product ideafs, proving, under suitable conditions, that the center of a Lie affgebra is an ideaf. We then develop the notion of affine nilpotency and establish connections between the nilpotency of Lie affgebras and that of their retracted Lie algebras. As an application, we prove an Engel-type theorem for Lie affgebras of the form $\mathfrak{a}(\mathfrak{g};\kappa=2\lambda,\lambda,s)$.

math.RA↗

arXiv2025

A General Characterization on the Uniqueness Problem of L-Functions and General Meromorphic Functions

In the paper, concerning a question of Yi [23], we study general criterion for the uniqueness of an L-function and a general meromorphic function. Our results improve and extend all the existing results in this direction [23, 18, 17, 4] to the most general setting. Moreover, we have exhibited a handsome number of examples to justify our claims as well as to confirm the wide-ranging applications of our results.

math.CV↗

arXiv2025

On Affine Version of Hom-Lie Algebras

This paper introduces Hom-type analogues of affine algebraic structures, termed Hom-affgebras. Extending Brzezi\'nski's theory of affgebras and the Hom-algebra framework developed by Hartwig-Larsson-Silvestrov, we define and study Hom-associative, Hom-pre-Lie, and Hom-Lie affgebras, where the classical identities are twisted by an affine self-map. We show how Hom-associative, Hom-pre-Lie, and Hom-Lie affgebras are related to one another. The main focus of this paper is on Hom-Lie affgebras and their fibers. We study the concept of generalized derivations for Hom-Lie algebras, extending the notion of generalized derivations for Lie algebras. We explore the close relationship between Hom-Lie affgebras and such derivations. We show that every Hom-Lie affgebra both determines and is determined by a Hom-Lie algebra together with such a generalized derivation and a constant. Furthermore, we establish that a homomorphism between Lie affgebras corresponds to a homomorphism between their associated Lie fibers along with a constant, and vice versa.

math.RA↗

arXiv2025

On Hom-Analogues of Heaps and Trusses

This paper introduces Hom-heaps, Hom-trusses, and Hom-braces as Hom-type analogues of their classical counterparts. We establish the correspondence between Hom-heaps and Hom-groups by showing that the retract of a Hom-heap at a point forms a Hom-group precisely when the point is fixed by the twisting map, and prove that translation maps induce isomorphisms between Hom-group retracts at different fixed base points. We introduce three equivalent notions of Hom-trusses and investigate their structural properties. We also propose three variants of Hom-braces and establish their correspondence with Hom-trusses, showing that certain Hom-trusses naturally give rise to Hom-braces and conversely. These results provide a unified framework extending heap and truss theory to the Hom-algebraic setting, with potential applications to the Yang--Baxter equation and non-associative geometry.

math.RA↗

arXiv2025

Cohomology and Extensions of $C_p$-Green Functors of Lie Type

We develop a theory of $C_p$-Green functors of Lie type, unifying the axiomatic framework of Green functors with the structure of Lie algebras under the action of a cyclic group $C_p$ of prime order. Extending classical notions from representation theory and topology, we define tensor and exterior products, introduce an equivariant Chevalley-Eilenberg cohomology, and construct cup products that endow the cohomology with a graded Green functor of Lie type structure. A key result establishes a correspondence between equivalence classes of singular extensions and second cohomology groups, generalizing classical Lie algebra extension theory to the equivariant setting. This framework enriches the toolkit for studying equivariant algebraic structures and paves the way for further applications in deformation theory, homotopical algebra, and representation theory.

math.RA↗

arXiv2024

A cohomological study of modified Rota-Baxter associative algebras with derivations

This paper presents a cohomological study of modified Rota-Baxter associative algebras in the presence of derivations. The Modified Rota-Baxter operator, which is a modified version and closely related to the classical Rota-Baxter operator, has garnered significant attention due to its applications in various mathematical and physical contexts. In this study, we define a cohomology theory and also investigate a one-parameter formal deformation theory and abelian extensions of modified Rota-Baxter associative algebras under the influence of derivations.

math.RA↗

arXiv2024

On Nijenhuis Lie triple systems

In this paper, we investigate the mathematical structure of Nijenhuis Lie triple systems, an extension of classical Lie triple systems augmented with the Nijenhuis operator. Our study focuses on the cohomology of Nijenhuis Lie triple systems and demonstrates how abelian extensions of Nijenhuis Lie triple systems are related to cohomology groups. Additionally, we define Nijenhuis Lie triple 2-systems and also classify `strict' and `skeletal' Nijenhuis Lie triple 2-systems in terms of crossed modules and the cohomology of Nijenhuis Lie triple systems.

math.RA↗

arXiv2023

Deformation cohomology of morphisms of Lie-Yamaguti algebras

We study cohomology of morphisms of Lie-Yamaguti algebras. As an application, we establish that this cohomology `controls' the formal deformations. Additionally, we demonstrate its connection to the abelian extension of morphisms of Lie-Yamaguti algebras.

math.RA↗

arXiv2023

Cohomology of modified Rota-Baxter Leibniz algebra of weight $λ$

Rota-Baxter operators have been paid much attention in the last few decades as they have many applications in mathematics and physics. In this paper, our object of study is modified Rota-Baxter operators on Leibniz algebras. We investigate modified Rota-Baxter Leibniz algebras from the cohomological point of view. We study a one-parameter formal deformation theory of modified Rota-Baxter Leibniz algebras and define the associated deformation cohomology that controls the deformation. Finally, as an application, we characterize equivalence classes of abelian extensions in terms of second cohomology groups.

math.RA↗

arXiv2023

On $α$-type (equivariant) cohomology of Hom-pre-Lie algebras

In this paper, we define a new cohomology theory for multiplicative Hom-pre-Lie algebras which controls deformations of Hom-pre-Lie algebra structure. This new cohomology is a natural one by considering the structure map. We develop equivariant cohomology theory for a Hom-pre-Lie algebra equipped with a finite group action by formulating a proper notion of coefficients system for the equivariant cohomology. We also study the associated formal deformation theory for Hom-pre-Lie algebras in the equivariant context.

math.RA↗

arXiv2023

Nijenhuis operators on Leibniz algebras

In this paper, we study Nijenhuis operators on Leibniz algebras. We discuss the relationship of Nijenhuis operators with Rota-Baxter operators and modified Rota-Baxter operators on Leibniz algebras. We define a representation theory of Nijenhuis Leibniz algebras and construct a cohomology theory. Next, we define a one-parameter formal deformation theory of Nijenhuis Leibniz algebras and study infinitesimals, rigidity, and equivalences along the line of Gerstenhaber deformation theory. As an application of our cohomology theory, we show that our cohomology is deformation cohomology and study abelian extensions of such algebras.

math.RA↗

arXiv2023

On compatible Leibniz algebras

In this paper, we study compatible Leibniz algebras. We characterize compatible Leibniz algebras in terms of Maurer-Cartan elements of a suitable differential graded Lie algebra. We define a cohomology theory of compatible Leibniz algebras which in particular controls a one-parameter formal deformation theory of this algebraic structure. Motivated by a classical application of cohomology, we moreover study the abelian extension of compatible Leibniz algebras.

math.RA↗

arXiv2023

Cohomology, deformations and extensions of Rota-Baxter Leibniz algebras

A Rota-Baxter Leibniz algebra is a Leibniz algebra $(\mathfrak{g},[~,~]_{\mathfrak{g}})$ equipped with a Rota-Baxter operator $T : \mathfrak{g} \rightarrow \mathfrak{g}$. We define representation and dual representation of Rota-Baxter Leibniz algebras. Next, we define a cohomology theory of Rota-Baxter Leibniz algebras. We also study the infinitesimal and formal deformation theory of Rota-Baxter Leibniz algebras and show that our cohomology is deformation cohomology. Moreover, We define an abelian extension of Rota-Baxter Leibniz algebras and show that equivalence classes of such extensions are related to the cohomology groups.

math.RA↗

arXiv2022

On deformation cohomology of compatible Hom-associative algebras

In this paper, we consider compatible Hom-associative algebras as a twisted version of compatible associative algebras. Compatible Hom-associative algebras are characterized as Maurer-Cartan elements in a suitable bidifferential graded Lie algebra. We also define a cohomology theory for compatible Hom-associative algebras generalizing the classical case. As applications of cohomology, we study abelian extensions and deformations of compatible Hom-associative algebras.

math.RA↗

arXiv2022

A Fractional Image Inpainting Model Using a Variant of Mumford-Shah Model

In this paper, we propose a fourth order PDE model for image inpainting based on a variant of the famous Mumford-Shah (MS) image segmentation model. Convexity splitting is used to discrtised the time and we replace the Laplacian by its fractional counterpart in the time discretised scheme. Fourier spectral method is used for space discretization. Consistency, stability and convergence of the time discretised model has been carried out. The model is tested on some standard test images and compared them with the result of some models existed in the literature.

math.NA↗

arXiv2022

On equivariant Lie-Yamaguti algebras and related structures

In this paper, we first discuss cohomology and a one-parameter formal deformation theory of Lie-Yamaguti algebras. Next, we study finite group actions on Lie-Yamaguti algebras and introduce equivariant cohomology for Lie-Yamaguti algebras equipped with group actions. Finally, we study an equivariant one-parameter formal deformation theory and show that our equivariant cohomology is the suitable deformation cohomology.

math.RA↗

arXiv2021

On 3-Lie algebras with a derivation

In this paper, we study 3-Lie algebras with derivations. We call the pair consisting of a 3-Lie algebra and a distinguished derivation by the 3-LieDer pair. We define a cohomology theory for 3-LieDer pair with coefficients in a representation. We study central extensions of a 3-LieDer pair and show that central extensions are classified by the second cohomology of the 3-LieDer pair with coefficients in the trivial representation. We generalize Gerstenhaber's formal deformation theory to 3-LieDer pairs in which we deform both the 3-Lie bracket and the distinguished derivation.

math.RA↗

arXiv2021

Rota-Baxter operators and related structures on anti-flexible algebras

In this paper, we first construct a graded Lie algebra which characterizes Rota-Baxter operators on an anti-flexible algebra as Maurer-Cartan elements. Next, we study infinitesimal deformations of bimodules over anti-flexible algebras. We also consider compatible Rota-Baxter operators on bimodules over anti-flexible algebras. Finally, We define $\mathcal{ON}$-structures which give rise to compatible Rota-Baxter operators and vice-versa.

math.RA↗