Searcharxiv⌕ Search

arXiv subjects

Goutam Mukherjee

Publications and source records attributed to Goutam Mukherjee.

At least 19 recordsLinked to original sources
arXiv2025

Universal Coacting Hopf algebra of a Finite dimensional Lie-Yamaguti algebra

M. E. Sweedler first constructed a universal Hopf algebra of an algebra. It is known that the dual notions to the existing ones play a dominant role in Hopf algebra theory. Yu. I. Manin and D. Tambara introduced the dual notion of Sweedler's construction in separate works. In this paper, we construct a universal algebra for a finite-dimensional Lie-Yamaguti algebra. We demonstrate that this universal algebra possesses a bialgebra structure, leading to a universal coacting Hopf algebra for a finite-dimensional Lie-Yamaguti algebra. Additionally, we develop a representation-theoretic version of our results. As an application, we characterize the automorphism group and classify all abelian group gradings of a finite-dimensional Lie-Yamaguti algebra.

math.RA↗
arXiv2025

Lie-Yamaguti Algebra Bundle

We introduce the notion of Lie-Yamaguti algebra bundle, define its cohomology groups with coefficients in a representation and show that such bundles appeared naturally from geometric considerations in the work of M. Kikkawa, which motivates us to introduce this object in the proper mathematical framework. We also study abelian extensions of Lie-Yamaguti algebra bundles and investigate their relationship with suitable cohomology group.

math.RA↗
arXiv2023

Poisson Dialgebras

The notion of Poisson dialgebras was introduced by Loday. In this article, we propose a new definition with some modifications that is supported by several canonical examples coming from Poisson algebra modules, averaging operators on Poisson algebras, and differential Poisson algebras. We show that a Poisson object in the category of linear maps has an associated Poisson dialgebra structure. Conversely, starting from a Poisson dialgebra we describe a Poisson object in the category of linear maps. These constructions yield a pair of adjoint functors between the category of Poisson objects in the category of linear maps and the category of Poisson dialgebras. There is a Lie $2$-algebra associated with any Leibniz algebra. Here, we first obtain an associative $2$-algebra starting from a dialgebra. Then, for a Poisson dialgebra, we construct a graded space that inherits both a Lie $2$-algebra and an associative $2$-algebra structure. In a particular case of Poisson dialgebras, which we call `reduced Poisson dialgebra', we obtain an associated $2$-term homotopy Poisson algebra (of degree $0$).

math.RA↗
arXiv2023

Automorphisms of extensions of Lie-Yamaguti algebras and Inducibility problem

Lie-Yamaguti algebras generalize both the notions of Lie algebras and Lie triple systems. In this paper, we consider the inducibility problem for automorphisms of extensions of Lie-Yamaguti algebras. More precisely, given an abelian extension $$0 \to V \xrightarrow[]{i} \widetilde{L} \xrightarrow[]{p} L \to 0$$ of a Lie-Yamaguti algebra $L$, we are interested in finding the pairs $(ϕ, ψ)\in \mathrm{Aut}(V)\times \mathrm{Aut}(L)$, which are inducible by an automorphism in $\mathrm{Aut}(\widetilde{L})$. We connect the inducibility problem to the $(2,3)$-cohomology of Lie-Yamaguti algebra. In particular, we show that the obstruction for a pair of automorphism in $\mathrm{Aut}(V)\times \mathrm{Aut}(L)$ to be inducible lies in the $(2,3)$-cohomology group $\mathrm{H}^{(2,3)}(L,V)$. We develop the Wells exact sequence for Lie-Yamaguti algebra extensions, which relates the space of derivations, automorphism groups, and $(2,3)$-cohomology groups of Lie-Yamaguti algebras. As an application, we describe certain automorphism groups of semi-direct product Lie-Yamaguti algebras. In the sequel, we apply our results to discuss inducibility problem for nilpotent Lie-Yamaguti algebras of index $2$. We give examples of infinite families of such nilpotent Lie-Yamaguti algebras and characterize the inducible pairs of automorphisms for extensions arising from these examples. Finally, we write an algorithm to find out all the inducible pairs of automorphisms for extensions arising from nilpotent Lie-Yamaguti algebras of index $2$.

math.RA↗
arXiv2022

A note on cohomology of Clifford algebras

In this article we construct a cochain complex of a complex Clifford algebra with coefficients in itself in a combinatorial fashion and we call the corresponding cohomology by {\it Clifford cohomology.} We show that {\it Clifford cohomology} controls the deformation of a complex Clifford algebra and can classify them up to Morita equivalence. We also study Hochschild cohomology groups and formal deformations of the algebra of smooth sections of a complex Clifford algebra bundle over an even dimensional orientable Riemannian manifold \(M\) which admits a \(Spin^{c}\) structure.

math.AT↗
arXiv2021

Cohomology and deformations of Filippov algebroids

In this article, we study the deformations of Filippov algebroids. We define a differential graded Lie algebra (in short DGLA) for a Filippov algebroid by introducing the notion of Filippov multiderivations for a vector bundle. Later on, we discuss deformations of a Filippov algebroid in terms of low-dimensional cohomology associated to this DGLA. We define Nijenhuis operators on Filippov algebroids and characterize trivial deformations of Filippov algebroids in terms of these operators. In the end, we define finite order deformations and discuss the problem of extending a given finite order deformation to a deformation of a higher order.

math.DG↗
arXiv2020

Equivariant one-parameter formal deformations of Hom-Leibniz algebras

Aim of this paper is to define a new type of cohomology for multiplicative Hom-Leibniz algebras which controls deformations of Hom-Leibniz algebra structure. The cohomology and the associated deformation theory for Hom-Leibniz algebras as developed here are also extended to equvariant context, under the presence of finite group actions on Hom-Leibniz algebras.

math.RA↗
arXiv2018

Nambu Structures And Associated Bialgebroids

This paper investigates higher order generalizations of well known results for Lie algebroids and bialgebroids. It is proved that $n$-Lie algebroid structures correspond to $n$-ary generalization of Gerstenhaber algebras and are implied by $n$-ary generalization of linear Poisson structures on the dual bundle. A Nambu-Poisson manifold (of order $n>2$) gives rise to a special bialgebroid structure which is referred to as a weak Lie-Filippov bialgebroid (of order $n$). It is further demonstrated that such bialgebroids canonically induce a Nambu-Poisson structure on the base manifold. Finally, the tangent space of a Nambu Lie group gives an example of a weak Lie-Filippov bialgebroid over a point.

math.DG↗
arXiv2017

Nambu structures on Lie algebroids and their modular classes

We introduce the notion of the modular class of a Lie algebroid equipped with a Nambu structure. In particular, we recover the modular class of a Nambu-Poisson manifold $M$ with its Nambu tensor $Λ$ as the modular class of the tangent Lie algebroid $TM$ with Nambu structure $Λ.$ We show that many known properties of the modular class of a Nambu-Poisson manifold that of a Lie algebropid extend to the setting of a Lie algebroid with Nambu structure. Finally, we prove that for a large class of Nambu-Poisson manifolds considered as tangent Lie algebroids with Nambu structures, the associated modular classes are closely related to Evens-Lu-Weinstein modular classes of Lie algebroids.

math.DG↗
arXiv2014

Modular Class of a Lie algebroid with a Nambu structure

In this paper, we introduce the notion of modular class of a Lie algebroid $A$ equipped with a Nambu structure satisfying some suitable hypothesis. We also introduce cohomology and homology theories for such Lie algebroids and prove that these theories are connected by a duality isomorphism when the modular class is null.

math.DG↗
arXiv2013

Versal deformation theory of algebras over a quadratic operad

We develop deformation theory of algebras over quadratic operads where the parameter space is a commutative local algebra. We also give a construction of a distinguised deformation of an algebra over a quadratic operad with a complete local algebra as its base--the so called `versal deformation'--which induces all other deformations of the given algebra.

math.KT↗
arXiv2013

Some Computations in Equivariant cobordism in relation to Milnor manifolds

Let $\mathcal{N}_*$ be the unoriented cobordism algebra, let $G=(\Z_2)^n$ and let $Z_*(G)$ denote the equivariant cobordism algebra of $G$-manifolds with finite stationary point sets. Let $ε_* :Z_*(G) \to \mathcal{N}_*$ be the homomorphism which forgets the $G$-action. We use Milnor manifolds (degree 1 hypersurfaces in $\R P^m\times \R P^n$) to construct non-trivial elements in $Z_*(G)$. We prove that these elements give rise to indecomposable elements in $Z_*(G)$ in degrees up to $2^n - 5$. Moreover, in most cases these elements can be arranged to be in $\mathit{Ker}(ε_*)$.

math.AT↗
arXiv2013

Finite group actions on Kan complexes

We study simplicial action of groups on one vertex Kan complexes. We show that every semi-direct product of the fundamental group of an one vertex Kan complex with a finite group can be simplicially realized. We also calculate the cohomology of the fixed point set of a finite $p-$group action on an one vertex aspherical Kan complex.

math.AT↗
arXiv2010

Equivariant Simplicial Cohomology With Local Coefficients and its Classification

We introduce equivariant twisted cohomology of a simplicial set equipped with simplicial action of a discrete group and prove that for suitable twisting function induced from a given equivariant local coefficients, the simplicial version of Bredon-Illman cohomology with local coefficients is isomorphic to equivariant twisted cohomology. The main aim of this paper is to prove a classification theorem for equivariant simplicial cohomology with local coefficients.

math.AT↗