arXiv subjects

Shuangjian Guo

Publications and source records attributed to Shuangjian Guo.

At least 37 records · Page 2Linked to original sources

arXiv2021

Relative Rota-Baxter systems on Leibniz algebras

In this paper, we introduce relative Rota-Baxter systems on Leibniz algebras and give some characterizations and new constructions. Then we construct a graded Lie algebra whose Maurer-Cartan elements are relative Rota-Baxter systems. This allows us to define a cohomology theory associated with a relative Rota-Baxter system. Finally, we study formal deformations and extendibility of finite order deformations of a relative Rota-Baxter system in terms of the cohomology theory.

math.RA↗

arXiv2020

The Hom-Long dimodule category and nonlinear equations

In this paper, we construct a kind of new braided monoidal category over two Hom-Hopf algerbas $(H,α)$ and $(B,β)$ and associate it with two nonlinear equations. We first introduce the notion of an $(H,B)$-Hom-Long dimodule and show that the Hom-Long dimodule category $^{B}_{H} \Bbb L$ is an autonomous category. Second, we prove that the category $^{B}_{H} \Bbb L$ is a braided monoidal category if $(H,α)$ is quasitriangular and $(B,β)$ is coquasitriangular and get a solution of the quantum Yang-Baxter equation. Also, we show that the category $^{B}_{H} \Bbb L$ can be viewed as a subcategory of the Hom-Yetter-Drinfeld category $^{HøB}_{HøB} \Bbb {HYD}$. Finally, we obtain a solution of the Hom-Long equation from the Hom-Long dimodules.

math.RA↗

arXiv2020

$n$-dervations of Lie color algebras

The aim of this article is to discuss the $n$-derivation algebras of Lie color algebras. It is proved that, if the base ring contains $\frac{1}{n-1}$, $L$ is a perfect Lie color algebra with zero center, then every triple derivation of $L$ is a derivation, and every $n$-derivation of the derivation algebra $nDer(L))$ is an inner derivation.

math.RA↗

arXiv2020

Crossed product Hom-Hopf algebras and lazy 2-cocycle

Let $(H,\a)$ be a Hom-Hopf algebra and $(A,\b)$ be a Hom-algebra. In this paper we will construct the Hom-crossed product $(A\#_σH,\bø\a)$, and prove that the extension $A\subseteq A\#_σH$ is actually a Hom-type cleft extension and vice versa. Then we will give the necessary and sufficient conditions to make $(A\#_σH,\bø\a)$ a Hom-Hopf algebra. Finally we will study the lazy 2-cocycle on $(H,\a)$.

math.RA↗

arXiv2020

On split regular Hom-Leibniz-Rinehart algebras

In this paper, we introduce the notion of the Hom-Leibniz-Rinehart algebra as an algebraic analogue of Hom-Leibniz algebroid, and prove that such an arbitrary split regular Hom-Leibniz-Rinehart algebra $L$ is of the form $L=U+\sum_γI_γ$ with $U$ a subspace of a maximal abelian subalgebra $H$ and any $I_γ$, a well described ideal of $L$, satisfying $[I_γ, I_δ]= 0$ if $[γ]\neq [δ]$. In the sequel, we develop techniques of connections of roots and weights for split Hom-Leibniz-Rinehart algebras respectively. Finally, we study the structures of tight split regular Hom-Leibniz-Rinehart algebras.

math.RA↗

arXiv2019

Representations and deformations of Hom-Lie-Yamaguti superalgebras

Let $(L, α)$ be a Hom-Lie-Yamaguti superalgebra. We first introduce the representation and cohomology theory of Hom-Lie-Yamaguti superalgebras. Furthermore, we introduce the notions of generalized derivations and representations of $(L, α)$ and present some properties. Finally, we investigate the deformations of $(L, α)$ by choosing some suitable cohomology.

math.RA↗

arXiv2019

On 3-Hom-Lie-Rinehart algebras

We introduce the notion of 3-Hom-Lie-Rinehart algebra and systematically describe a cohomology complex by considering coefficient modules. Furthermore, we consider extensions of a 3-Hom-Lie-Rinehart algebra and characterize the first cohomology space in terms of the group of automorphisms of an $A$-split abelian extension and the equivalence classes of $A$-split abelian extensions. Finally, we study formal deformations of 3-Hom-Lie-Rinehart algebras.

math.RA↗

arXiv2019

On the structure of split regular Hom-Lie Rinehart algebras

The aim of this paper is to study the structures of split regular Hom-Lie Rinehart algebras. Let $(L,A)$ be a split regular Hom-Lie Rinehart algebra. We first show that $L$ is of the form $L=U+\sum_{[γ]\inΓ/\thicksim}I_{[γ]}$ with $U$ a vector space complement in $H$ and $I_{[γ]}$ are well described ideals of $L $ satisfying $I_{[γ]},I_{[δ]}=0$ if $I_{[γ]}\neq I_{[δ]}$. Also, we discuss the weight spaces and decompositions of $A$ and present the relation between the decompositions of $L$ and $A$. Finally, we consider the structures of tight split regular Hom-Lie Rinehart algebras.

math.RA↗

arXiv2019

On split regular BiHom-Leibniz superalgebras

The goal of this paper is to study the structure of split regular BiHom-Leiniz superalgebras, which is a natural generalization of split regular Hom-Leiniz algebras and split regular BiHom-Lie superalgebras. By developing techniques of connections of roots for this kind of algebras, we show that such a split regular BiHom-Leiniz superalgebras $\mathfrak{L}$ is of the form $\mathfrak{L}=U+\sum_{\a}I_\a$ with $U$ a subspace of a maximal abelian subalgebra $H$ and any $I_{\a}$, a well described ideal of $\mathfrak{L}$, satisfying $[I_\a, I_\b]= 0$ if $[\a]\neq [\b]$. In the case of $\mathfrak{L}$ being of maximal length, the simplicity of $\mathfrak{L}$ is also characterized in terms of connections of roots.

math.RA↗

arXiv2019

Derivations and deformations of $δ$-Jordan Lie supertriple systems

Let $T$ be a $δ$-Jordan Lie supertriple system. We first introduce the notions of generalized derivations and representations of $T$ and present some properties. Also, we study the low dimension cohomology and the coboundary operator of $T$, and then we investigate the deformations and Nijenhuis operators of $T$ by choosing some suitable cohomology.

math.RA↗

arXiv2019

Central invariants and enveloping algebras of braided Hom-Lie algebras

Let $(H,α)$ be a monoidal Hom-Hopf algebra and $^{H}_{H}\mathcal{HYD}$ the Hom-Yetter-Drinfeld category over $(H,α)$. Then in this paper, we first introduce the definition of braided Hom-Lie algebras and show that each monoidal Hom-algebra in $^{H}_{H}\mathcal{HYD}$ gives rise to a braided Hom-Lie algebra. Second, we prove that if $(A,β)$ is a sum of two $H$-commutative monoidal Hom-subalgebras, then the commutator Hom-ideal $[A,A]$ of $A$ is nilpotent. Also, we study the central invariant of braided Hom-Lie algebras as a generalization of generalized Lie algebras. Finally, we obtain a construction of the enveloping algebras of braided Hom-Lie algebras and show that the enveloping algebras are $H$-cocommutative Hom-Hopf algerbas.

math.RA↗

arXiv2019

On split regular BiHom-Poisson superalgebras

The paper introduces the class of split regular BiHom-Poisson superalgebras, which is a natural generalization of split regular Hom-Poisson algebras and split regular BiHom-Lie superalgebras. By developing techniques of connections of roots for this kind of algebras, we show that such a split regular BiHom-Poisson superalgebras $A$ is of the form $A=U+\sum_{\a}I_\a$ with $U$ a subspace of a maximal abelian subalgebra $H$ and any $I_{\a}$, a well described ideal of $A$, satisfying $[I_\a, I_\b]+I_\a I_\b = 0$ if $[\a]\neq [\b]$. Under certain conditions, in the case of $A$ being of maximal length, the simplicity of the algebra is characterized.

math.RA↗

arXiv2019

The constructions of 3-Hom-Lie bialgebras

In this paper, we first introduce the notion of a 3-Hom-Lie bialgebra and prove that it is equivalent to a Manin triple of 3-Hom-Lie algebras. Also, we study the $\mathcal{O}$-operator and construct solutions of the 3-Lie classical Hom-Yang-Baxter equation interms of $\mathcal{O}$-operators and 3-Hom-pre-Lie algebras. Finally, we show that a 3-Hom-Lie algebra has a phase space if and only if it is sub-adjacent to a 3-Hom-pre-Lie algebra.

math.RA↗

arXiv2018

Some results on $δ$-Hom-Jordan Lie conformal superalgebras

In this paper, we introduce the representation theory of $δ$-Hom-Jordan Lie conformal superalgebras and discuss the cases of adjoint representations. Furthermore, we develop the cohomology theory of Hom-Lie conformal superalgebras and discuss some applications to the study of deformations of regular $δ$-Hom-Jordan Lie conformal superalgebras. Finally, we introduce derivations of multiplicative $δ$-Hom-Jordan Lie conformal superalgebras and study their properties.

math.RA↗

arXiv2018

Cohomology and derivations of BiHom-Lie conformal algebras

In this paper, we introduce the notion of a BiHom-Lie conformal algebra and develop its cohomology theory. Also, we discuss some applications to the study of deformations of regular BiHom-Lie conformal algebras. Finally, we introduce derivations of multiplicative BiHom-Lie conformal algebras and study their properties.

math.RA↗

arXiv2018

The construction of Hom left-symmetric conformal bialgebras

In this paper, we first introduce the notion of Hom-left-symmetric conformal bialgebras and show some nontrivial examples. Also, we present construction methods of matched pairs of Hom-Lie conformal algebras and Hom-left-symmetric conformal algebras. Finally, we prove that a finite Hom-left-symmetric conformal bialgebra is free as a $\mathbb{C}[\partial]$-module is equivalent to a Hom-parakähler Lie conformal algebra. In particular, we investigate the coboundary Hom-left-symmetric conformal bialgebras.

math.RA↗

arXiv2018

Manin triples and quasitriangular structures of Hom-Poisson bialgebras

In this paper, we first introduce the definition of a Hom-Poisson bialgebra and give an equivalent descriptions via the Manin triple of Hom-Poisson algebras. Also we introduce notions of $\mathcal{O}$-operator on a Hom-Poisson algebra, post-Hom-Poisson algebra and quasitriangular Hom-Poisson bialgebra, and present a method to construct post-Hom-Poisson algebras. Finally, we show that a quasitriangular Hom-Poisson bialgebra naturally yields a post-Hom-Poisson algebra.

math.RA↗

arXiv2018

Representations and derivations of Hom-Lie conformal superalgebras

In this paper, we introduce a representation theory of Hom-Lie conformal superalgebras and discuss the cases of adjoint representations. Furthermore, we develop cohomology theory of Hom-Lie conformal superalgebras and discuss some applications to the study of deformations of regular Hom-Lie conformal superalgebras. Finally, we introduce derivations of multiplicative Hom-Lie conformal superalgebras and study their properties.

math.RA↗