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Shuangjian Guo

Publications and source records attributed to Shuangjian Guo.

At least 19 recordsLinked to original sources
arXiv2026

Modified Rota-Baxter operators of non-zero weight on $3$-Lie algebras

In this paper, we introduce the notion of modified Rota-Baxter operators of non-zero weight on $3$-Lie algebras and provide some examples. Next, we give various constructions of modified Rota-Baxter operators of non-zero weight according to constructions of $3$-Lie algebras. Furthermore, we define a cohomology of modified Rota-Baxter operators of non-zero weight on $3$-Lie algebras with coefficients in a suitable representation. As an application, we study formal deformations of modified Rota-Baxter operators of non-zero weight that are generated by the above-defined cohomology. In the final part of the paper, we construct two \(L_\infty[1]\)-algebra structures whose Maurer-Cartan elements correspond to relative and absolute modified Rota-Baxter \(3\)-Lie algebra structures of nonzero weight, respectively. Lastly, we compare our \(L_\infty[1]\)-algebraic approach with the deformation-controlling \(L_\infty[1]\)-algebra for relative Rota-Baxter \(3\)-Lie operators developed by Hou, Sheng, and Zhou.

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arXiv2025

Leibniz-dendriform bialgebras and relative Rota-Baxter operators

In this paper, we introduce the notion of Leibniz-dendriform bialgebras and establish their equivalence with phase spaces and matched pairs of Leibniz algebras. The study of the coboundary case leads naturally to the Leibniz-dendriform Yang-Baxter equation (LD-YBE). We prove that skew-symmetric solutions of the LD-YBE give rise to coboundary Leibniz-dendriform bialgebras. Furthermore, we demonstrate that solutions not necessarily skew-symmetric can also induce such bialgebras. This observation motivates the introduction of quasi-triangular and factorizable Leibniz-dendriform bialgebras. In particular, we show that solutions of the LD-YBE with invariant symmetric parts yield quasi-triangular Leibniz-dendriform bialgebras. Such solutions are also interpreted as relative Rota-Baxter operators with weights. Finally, we establish a one-to-one correspondence between quadratic Rota-Baxter Leibniz-dendriform algebras and factorizable Leibniz-dendriform bialgebras.

math.RA↗
arXiv2025

Embedding tensors on 3-Leibniz algebras and their derived algebraic structures and deformations

In this paper, first we introduce the notions of 3-tri-Leibniz algebras and embedding tensors on 3-Leibniz algebras. We show that an embedding tensor gives rise to a 3-tri-Leibniz algebra. Conversely, a 3-tri-Leibniz algebra gives rise to a 3-Leibniz algebra and a representation such that the quotient map is an embedding tensor. Furthermore, any 3-tri-Leibniz algebra can be embedded into an averaging 3-Leibniz algebra. Next, we introduce the notion of 3-tri-Leibniz dialgebras and demonstrate that homomorphic embedding tensors inherently induce 3-tri-Leibniz dialgebras. Finally, we study the linear deformations of embedding tensors by defining first cohomology.

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arXiv2024

Non-abelian extensions of Lie triple systems and Wells exact sequences

In this paper, we investigate non-abelian extensions and inducibility of pairs of automorphisms of Lie triple systems. First, we introduce non-abelian cohomology groups and classify the non-abelian extensions in terms of non-abelian cohomology groups. Next, we characterize the non-abelian extensions using Maurer-Cartan elements. Furthermore, we explore the inducibility of pairs of automorphisms and derive the analog Wells exact sequences under the circumstance of Lie triple systems. Finally, we state the previous results under the context of abelian extensions of Lie triple systems.

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arXiv2024

Cohomologies and deformations of weighted Rota-Baxter Lie algebras and associative algebras with derivations

The purpose of the present paper is to investigate cohomologies and deformations of weighted Rota-Baxter Lie algebras as well as weighted Rota-Baxter associative algebras with derivations. First we introduce a notion of weighted Rota-Baxter LieDer and weighted Rota-Baxter AssDer pairs. Then we construct cohomologies of weighted Rota-Baxter LieDer pairs, weighted Rota-Baxter AssDer pairs and we discuss the relation between their cohmologies. Finally, as an application, we study deformations of both of them.

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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.

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arXiv2024

Cohomologies of Reynolds Lie-Yamaguti algebras of any weight and applications

The purpose of the present paper is to investigate cohomologies of Reynolds Lie-Yamaguti algebras of any weight and provide some applications. First, we introduce the notion of Reynolds Lie-Yamaguti algebras and give some new examples. Moreover, cohomologies of Reynolds operators and Reynolds Lie-Yamaguti algebras with coefficients in a suitable representation are established. Finally, formal deformations and abelian extensions of Reynolds Lie-Yamaguti algebras are characterized in terms of lower degree cohomology groups.

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arXiv2024

Modified Rota-Baxter Lie-Yamaguti algebras

In this paper, first we introduce the concept of modified Rota-Baxter Lie-Yamaguti algebras. Then the cohomology of a modified Rota-Baxter Lie-Yamaguti algebra with coefficients in a suitable representation is established. As applications, the formal deformations and abelian extensions of modified Rota-Baxter Lie-Yamaguti algebras are studied using the second cohomology group.

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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.

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arXiv2022

Cohomology theory of Rota-Baxter pre-Lie algebras of arbitrary weights

This paper is devoted to studying deformation, cohomology theory of Rota-Baxter pre-Lie algebras of arbitrary weights. First we give the notion of a new representation of a Rota-Baxter pre-Lie algebra of arbitrary weight and define the cohomology theory of a Rota-Baxter pre-Lie algebra of arbitrary weight. Then we study formal deformations by lower degree cohomology groups. Finally, we classify abelian extensions of Rota-Baxter pre-Lie algebras of arbitrary weight using the second cohomology group, and classify skeletal Rota-Baxter pre-Lie 2-algebra of arbitrary weight using the third cohomology group as applications.

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arXiv2022

$L_\infty$-structures and cohomology theory of compatible $\mathcal {O}$-operators and compatible dendriform algebras

The notion of $\mathcal{O}$-operator is a generalization of the Rota-Baxter operator in the presence of a bimodule over an associative algebra. A compatible $\mathcal{O}$-operator is a pair consisting of two $\mathcal{O}$-operators satisfying a compatibility relation. A compatible $\mathcal{O}$-operator algebra is an algebra together with a bimodule and a compatible $\mathcal{O}$-operator. In this paper, we construct a graded Lie algebra and an $L_\infty$-algebra that respectively characterize compatible $\mathcal{O}$-operators and compatible $\mathcal{O}$-operator algebras as Maurer-Cartan elements. Using these characterizations, we define cohomology of these structures and as applications, we study formal deformations of compatible $\mathcal{O}$-operators and compatible $\mathcal{O}$-operator algebras. Finally, we consider a brief cohomological study of compatible dendriform algebras and find their relationship with the cohomology of compatible associative algebras and compatible $\mathcal{O}$-operators.

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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.

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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.

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arXiv2021

Hom-Yang-Baxter equations and Hom-Yang-Baxter systems

In this paper, we mainly present some new solutions of the Hom-Yang-Baxter equation from Hom-algebras, Hom-coalgebras and Hom-Lie algebras, respectively. Also, we prove that these solutions are all self-inverse and give some examples. Finally, we introduce the notion of Hom-Yang-Baxter systems and obtain two kinds of Hom-Yang-Baxter systems.

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.

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arXiv2021

On BiHom-analogue of generalized Lie algebras

In this paper, we introduce the definition of generalized BiHom-Lie algebras and generalized BiHom-Lie admissible algebras in the category ${}_H{\mathcal M}$ of left modules for any quasitriangular Hopf algebra $(H, R) $. Also, we describe the BiHom-Lie ideal structures of the BiHom-associative algebras.

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arXiv2021

Twisted relative Rota-Baxter operators on Leibniz algebras and NS-Leibniz algebras

In this paper, we introduce twisted relative Rota-Baxter operators on a Leibniz algebra as a generalization of twisted Poisson structures. We define the cohomology of a twisted relative Rota-Baxter operator $K$ as the Loday-Pirashvili cohomology of a certain Leibniz algebra induced by $K$ with coefficients in a suitable representation. Then we consider formal deformations of twisted relative Rota-Baxter operators from cohomological points of view. Finally, we introduce and study NS-Leibniz algebras as the underlying structure of twisted relative Rota-Baxter operators.

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