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Sumit Kumar Upadhyay

Publications and source records attributed to Sumit Kumar Upadhyay.

At least 19 recordsLinked to original sources
arXiv2026

Representation and Extending structure of a Multiplicative Lie algebra

In this paper, we attempt to develop a general notion of representation theory for multiplicative Lie algebras. Consequently, we study equivalent, irreducible, and completely reducible representations. We also introduce the notion of the action of a multiplicative Lie algebra on a vector space and show that there is a one-to-one correspondence between the action and the representation. We also studied the extended structure of a multiplicative Lie algebra through a group.

math.RT↗
arXiv2026

Theory of extension of commutative poisson algebra

In this paper, we develop the extension theory of commutative Poisson algebras. Specifically, we define the second cohomology of a commutative Poisson algebra $K$ with coefficients in an abelian Poisson algebra $H$. We then establish a natural bijective correspondence between this cohomology group and the set of equivalence classes of a special type of extensions.

math.RA↗
arXiv2025

Algebraic cohomotopy groups, algebraic fundamental groups and stably free modules

In this article, we prove the algebraic counterpart of the topological results $H^1(S^1, \mathbb{Z}) \cong \mathbb{Z}$ and $H^1(S^2, \mathbb{Z}) \cong \{0\}$. We also see that a non-trivial element of the algebraic cohomotopy groups of certain rings associated with some known topological spaces provides examples of non-free stably free module of rank two over those rings.

math.GR↗
arXiv2025

Excision and idealization of a multiplicative Lie algebra

In this article, we introduce the concepts of excision and idealization for a multiplicative Lie algebra (also for a Lie algebra), which provides two new multiplicative Lie algebras (or Lie algebras) from a given multiplicative Lie algebra (or Lie algebra) and an ideal, under certain conditions. These concepts may assist in classifying all multiplicative Lie algebras (or Lie algebras) of a specified order (or dimension).

math.GR↗
arXiv2024

Multiplicative Lie algebra structure on nilpotent group of class $2$

This paper explores the properties of multiplicative Lie algebra structures on a nilpotent group of class $2$. We also present a method for determining a multiplicative Lie algebra structure on a group that serves as an extension of one Lie ring by another Lie ring such as a metacyclic group and a nilpotent group of class $2$.

math.GR↗
arXiv2024

Tensor square and isoclinic extensions of multiplicative Lie algebras

In this paper, we discuss the capable and isoclinic properties of the tensor square in the context of multiplicative Lie algebras. We also developed the concept of isoclinic extensions and proved several results for multiplicative Lie algebras. Consequently, we demonstrate that covers of a multiplicative Lie algebra are mutually isoclinic.

math.GR↗
arXiv2024

Lie nilpotency and Lie solvability of tensor product of multiplicative Lie algebras

In this article, we discuss Lie nilpotency and Lie solvability of non-abelian tensor product of multiplicative Lie algebras. In particular, for giving information concerning the Lie nilpotency (or Lie solvability) of either multiplicative Lie algebras $G$ or $H,$ the non-abelian tensor product $ \frac{G\otimes H}{I}$ is Lie nilpotent (or Lie solvable), for some ideal $I$ of ${G\otimes H}$.

math.GR↗
arXiv2024

The Bogomolov multiplier of a multiplicative Lie algebra

In this paper, we develop the concept of the Bogomolov multiplier for a multiplicative Lie algebra and establish a Hopf-type formula. Consequently, we see that the Bogomolov multipliers of two isoclinic multiplicative Lie algebras are isomorphic.

math.GR↗
arXiv2023

Nilpotency and Capability in multiplicative Lie algebras

This paper aims to introduce the concept of nilpotency and capability in multiplicative Lie algebras. Also, we see the existence of covers of a multiplicative Lie algebra and thoroughly examine their relationships with capable and perfect multiplicative Lie algebras.

math.GR↗
arXiv2023

Isoclinism in multiplicative Lie algebras

The purpose of this paper is to introduce the notion of isoclinism and cover in a multiplicative Lie algebra which may be helpful to describe all multiplicative Lie algebra structures on a group. Consequently, we give the existence of the stem multiplicative Lie algebra. We also give the necessary and sufficient conditions for the existence of stem cover of a multiplicative Lie algebra.

math.RA↗
arXiv2022

The Schur multiplier of some finite multiplicative Lie algebras

The main aim of the article is to find the Schur multiplier and the Lie exterior square of some finite multiplicative Lie algebras. For a non abelian simple group $K$ with trivial Schur multiplier, we see that the Schur multiplier of multiplicative Lie algebra $K$ is trivial and the Lie exterior square of $K$ is an improper multiplicative Lie algebra $K.$

math.GR↗
arXiv2021

Gyrogroup extensions and semi cross product

The aim of the article is to study the extension theory of gyrogroups under certain conditions. Consequently, we see that there is an equivalence between the category GEXT of group-gyro extensions and the category GFAC of group-gyro factor systems. We also give a notion of semi cross product of a group and a gyrogroup which is a construction method of a larger class of gyrogroups.

math.GR↗
arXiv2021

Quillen Suslin theory for algebraic fundamental group

In this paper, we attempt to develop the Quillen Suslin theory for the algebraic fundamental group of a ring. We give a surjective group homomorphism from the algebraic fundamental group of the field of the real numbers to the group of integers. At the end of the paper, we also propose some problems related to the algebraic fundamental group of some particular type of rings.

math.GR↗
arXiv2021

Gyrogroup through its Grothendieck Group Completion and Right gyrogroup action

In this article, we discuss the Grothendieck group completion (GGC) of a gyrogroup. Consequently, we show that there is a one to one correspondence between actions and representations of a gyrogroup, and actions and representations of its Grothendieck group completion. We also introduce the concept of an action of a right gyrogroup.

math.GR↗
arXiv2019

Multiplicative Lie Algebra Structures on a Group

The main aim of this paper is to classify the distinct multiplicative Lie algebra structures (up to isomorphism) on a given group. We also see that for a given group $G$, every homomorphism from the non-abelian exterior square $G \wedge G$ to $G$ gives a multiplicative Lie algebra structure on $G$ under certain conditions.

math.GR↗
arXiv2018

A Construction of Bent Functions on a finite group

In this paper, we discuss when a class function on a finite group is a bent function. We have found a necessary condition for a class function on a finite abelian group to be bent. Also, we have found a necessary and sufficient condition for a class function on a finite cyclic group to be bent.

math.CO↗