arXiv subjects

Lamei Yuan

Publications and source records attributed to Lamei Yuan.

At least 19 recordsLinked to original sources

arXiv2026

Compatible Lie conformal bialgebras

We introduce and study compatible Lie conformal bialgebras as conformal counterparts of compatible Lie bialgebras. Such a structure consists of two compatible Lie conformal brackets and two compatible conformal cobrackets on the same $\C[\partial]$-module, and every simultaneous linear combination of them is again a Lie conformal bialgebra. We develop representations and matched pairs for compatible Lie conformal algebras, introduce compatible Lie conformal coalgebras, and establish their duality through the conformal dual in the finite case. For $\C[\p]$-modules that are free of finite rank, we prove the equivalence among compatible Lie conformal bialgebras, standard compatible conformal Manin triples and matched pairs. In the coboundary case, we characterize the tensors $r$ that determine compatible Lie conformal bialgebras. The characterization requires the symmetric part of $r$ to be invariant with respect to both brackets and three conformal Yang--Baxter conditions to hold. The first two conditions correspond to the two brackets separately, whereas the third is the compatible conformal Yang--Baxter condition. For comparison, we introduce the compatible conformal classical Yang--Baxter equation (CYBE) and show that each of its solutions satisfies these three conditions, while the converse fails. One explicit example shows that the first two conditions do not imply the third. Another shows that all three conformal Yang-Baxter conditions may hold even though $r$ is not a solution of compatible conformal CYBE.

math-ph↗

arXiv2026

On transposed Poisson conformal superalgebras

We introduce and study transposed Poisson conformal superalgebras, the $\mathbb Z_2$-graded conformal analogues of transposed Poisson algebras, as well as their noncommutative variants. We derive a family of identities forced by the transposed conformal super-Leibniz rule and prove that the tensor product over $\mathbb C[\partial]$ of two such superalgebras again carries a natural transposed Poisson conformal superalgebra structure. Moreover, we display a close relationship between transposed Poisson conformal superalgebras and Hom-Lie conformal superalgebras, and give the compatibility conditions between a Poisson conformal superalgebra and a transposed Poisson conformal superalgebra. In addition, several constructions are obtained from modified Lie conformal brackets and from Novikov-Poisson, pre-Lie commutative, differential Novikov-Poisson, and pre-Lie Poisson conformal superalgebras. Finally, using the known classification of Lie conformal superalgebras of rank (1+1), we determine all compatible transposed Poisson conformal superalgebra structures on such superalgebras.

math.RA↗

arXiv2026

Post-Lie conformal algebra structures on Lie conformal algebras

In this paper, we introduce and study post-Lie conformal algebras (PLCAs), a generalization of post-Lie algebras to conformal algebras. We establish an equivalence between PLCA structures and Rota-Baxter operators of weight 1 on Lie conformal algebras. We also show that every PLCA induces a new Lie conformal algebra and study PLCA structures on pairs of Lie conformal algebras. Finally, we classify all PLCA structures on two important classes of Lie conformal algebras: B(q) and W(b), achieved through Rota-Baxter operators of weight 1.

math.RA↗

arXiv2026

On transposed Poisson conformal algebras

The aim of this paper is to introduce the notion of (noncommutative) transposed Poisson conformal algebras, which serve as the conformal analogues of transposed Poisson algebras and admit a rich class of identities. We show that the tensor product of two transposed Poisson conformal algebras is also a transposed Poisson conformal algebra. Moreover, we establish a close relationship between transposed Poisson conformal algebras and Hom-Lie conformal algebras, and give the compatibility conditions between a Poisson conformal algebra and a transposed Poisson conformal algebra. In addition, we provide several constructions of transposed Poisson conformal algebras arising from related algebraic structures. Finally, a complete classification of compatible noncommutative transposed Poisson conformal algebraic structures over a class of Lie conformal algebras W(a, b) is given.

math.RA↗

arXiv2025

On L-dendriform conformal algebras

In this paper, we introduce the concept of L-dendriform conformal algebras, which arise naturally from the study of $\mathcal{O}$-operators on left-symmetric conformal algebras and solutions to the conformal $S$-equation. These algebras extend the classical notions of dendriform and left-symmetric conformal algebras, providing a unified algebraic framework for understanding compatible structures in conformal algebra theory. We establish fundamental properties of L-dendriform conformal algebras, explore their relationships with $\mathcal{O}$ -operators, Rota-Baxter operators, and Nijenhuis operators, and demonstrate their connections to dendriform and quadri conformal algebras. Additionally, we investigate compatible $\mathcal{O}$-operators and their induced compatible L-dendriform conformal algebra structures. Our results generalize and unify several existing algebraic structures in conformal algebra theory, offering new insights into their interplay and applications.

math.RA↗

arXiv2025

Computation of cohomology of finite Lie conformal algebras

By using the energy operator introduced by B. Bakalov, A. De Sole, and V.\,G. Kac, we propose an algorithm for computing the cohomology of finitely generated free Lie conformal algebras with a Virasoro element. This computational method enables us to determine cohomologies with coefficients in their conformal modules, including the trivial module, irreducible finitely free modules, and adjoint modules. As concrete applications, we compute by \textit{Mathematica} both basic and reduced cohomologies for the $\mathcal{W}(b)$, Schrödinger-Virasoro and extended Schrödinger-Virasoro Lie conformal algebras.

math.RA↗

arXiv2023

Cohomology of Twisted Rota-Baxter operators on Associative~Conformal Algebra

In this paper, we examine the concept of twisted Rota-Baxter (TRB) operators on associative conformal algebras. Our strategy begins by constructing an $L_\infty$-algebra using Maurer-Cartan elements derived from $H$-twisted Rota-Baxter ($H$-TRB) operators on associative conformal algebras. This structure leads us to explore the cohomology of the conformal $H$-TRB operator, which is characterized as the Hochschild cohomology of a specific associative conformal algebra with coefficients in a conformal bimodule. Furthermore, we study the linear and formal deformations of conformal $H$-TRB operators to explore the application of cohomology.

math.RA↗

arXiv2023

Twisting theory, relative Rota-Baxter type operators and $L_\infty$-algebras on Lie conformal algebras

Based on Nijenhuis-Richardson bracket and bidegree on the cohomology complex for a Lie conformal algebra, we develop a twisting theory of Lie conformal algebras. By using derived bracket constructions, we construct $L_\infty$-algebras from (quasi-)twilled Lie conformal algebras. And we show that the result of the twisting by a $\mathbb{C}[\partial]$-module homomorphism on a (quasi-)twilled Lie conformal algebra is also a (quasi-)twilled Lie conformal algebra if and only if the $\mathbb{C}[\partial]$-module homomorphism is a Maurer-Cartan element of the $L_\infty$-algebra. In particular, we show that relative Rota-Baxter type operators on Lie conformal algebras are Maurer-Cartan elements. Besides, we propose a new algebraic structure, called NS-Lie conformal algebras, that is closely related to twisted relative Rota-Baxter operators and Nijenhuis operators on Lie conformal algebras. As an application of twisting theory, we give the cohomology of twisted relative Rota-Baxter operators and study their deformations.

math.QA↗

arXiv2022

$\mathcal{O}$-operators and Nijenhius operators of associative conformal algebras

We study $\mathcal{O}$-operators of associative conformal algebras with respect to conformal bimodules. As natural generalizations of $\mathcal{O}$-operators and dendriform conformal algebras, we introduce the notions of twisted Rota-Baxter operators and conformal NS-algebras. We show that twisted Rota-Baxter operators give rise to conformal NS-algebras, the same as $\mathcal{O}$-operators induce dendriform conformal algebras. And we introduce a conformal analog of associative Nijenhius operators and enumerate main properties. By using derived bracket construction of Kosmann-Schwarzbach and a method of Uchino, we obtain a graded Lie algebra whose Maurer-Cartan elements are given by $\mathcal{O}$-operators. This allows us to construct cohomology of $\mathcal{O}$-operators. This cohomology can be seen as the Hochschild cohomology of an associative conformal algebra with coefficients in a suitable conformal bimodule.

math.RA↗

arXiv2022

Conformal $r$-matrix-Nijenhuis structures, symplectic-Nijenhuis structures and $\mathcal{O} N$-structures

In this paper, we first study infinitesimal deformations of a Lie conformal algebra and a Lie conformal algebra with a module (called an $\mathsf{LCMod}$ pair), which lead to the notions of Nijenhuis operator on the Lie conformal algebra and Nijenhuis structure on the $\mathsf{LCMod}$ pair, respectively. Then by adding compatibility conditions between Nijenhuis structures and $\mathcal{O}$-operators, we introduce the notion of an $\mathcal{O} N$-structure on an $\mathsf{LCMod}$ pair and show that an $\mathcal{O} N$-structure gives rise to a hierarchy of pairwise compatible $\mathcal{O}$-operators. In particular, we show that compatible $\mathcal{O}$-operators on a Lie conformal algebra can be characterized by Nijenhuis operators on Lie conformal algebras. Finally, we introduce the notions of conformal $r$-matrix-Nijenhuis structure and symplectic-Nijenhuis structure on the Lie conformal algebra and study their relations.

math.QA↗

arXiv2022

Biderivations of Hom-Lie algebras and superalgebras

On Hom-Lie algebras and superalgebras,we introduce the notions of biderivations, linear commuting maps and α-biderivations, and compute them for some typical Hom-Lie algebras and superalgebras, including q-deformed W(2,2) algebra, q-deformed Witt algebra and superalgebra.

math.RA↗

arXiv2019

Conformal modules and their extensions of a Lie conformal algebra related to a 2-dimensional Novikov algebra

Let $\mathcal{R}$ be a free Lie conformal algebra of rank $2$ with $\mathbb{C}[\partial]$-basis $\{L,I\}$ and relations \begin{eqnarray*} \left[L_λ L\right]=(\partial+2 λ) (L+I),\ \left[L_λ I\right]=(\partial+λ) I, \ \left[I_λ L\right]=λI,\ \left[I_λ I\right]=0. \end{eqnarray*} In this paper, we first classify all finite nontrivial irreducible conformal modules over $\mathcal{R}$. Then we determine extensions between two finite irreducible conformal $\mathcal{R}$-modules.

math.RT↗

arXiv2019

Extensions of Schrödinger-Virasoro conformal modules

In this paper, we study extensions between two finite irreducible conformal modules over the Schrödinger-Virasoro conformal algebra and the extended Schrödinger-Virasoro conformal algebra. Also, we classify all finite nontrivial irreducible conformal modules over the extended Schrödinger-Virasoro conformal algebra. As a byproduct, we obtain a classification of extensions of Heisenberg-Virasoro conformal modules.

math.RA↗

arXiv2019

Conformal biderivations of loop $W(a,b)$ Lie conformal algebra

In this paper, we study conformal biderivations of a Lie conformal algebra. First, we give the definition of conformal biderivation. Next, we determine the conformal biderivations of loop $W(a,b)$ Lie conformal algebra, loop Virasoro Lie conformal algebra and Virasoro Lie conformal algebra. Especially, all conformal biderivations on Virasoro Lie conformal algebra are inner conformal biderivations.

math.RA↗

arXiv2019

Unified products of Leibniz conformal algebras

The aim of this paper is to provide an answer to the $\mathbb{C}[\partial]$-split extending structures problem for Leibniz conformal algebras, which asks that how to describe all Leibniz conformal algebra structures on $E=R\oplus Q$ up to an isomorphism such that $R$ is a Leibniz conformal subalgebra. For this purpose, an unified product of Leibniz conformal algebras is introduced. Using this tool, two cohomological type objects are constructed to classify all such extending structures up to an isomorphism. Then this general theory is applied to the special case when $R$ is a free $\mathbb{C}[\partial]$-module and $Q$ is a free $\mathbb{C}[\partial]$-module of rank one. Finally, the twisted product, crossed product and bicrossed product between two Leibniz conformal algebras are introduced as special cases of the unified product, and some examples are given.

math.RA↗

arXiv2018

Extensions of modules over a class of Lie conformal algebras $\mathcal{W}(b)$

Let $\mathcal{W}(b)$ be a class of free Lie conformal algebras of rank $2$ with $\mathbb{C}[\partial]$-basis ${L,H}$ and relations \begin{eqnarray*} [L_λL]=(\partial+2λ)L,\ \ [L_λH]=\big(\partial+(1-b)λ\big)H, \ \ [H_λH]=0, \end{eqnarray*} where $b$ is a nonzero complex number. In this paper, we classify extensions between two finite irreducible conformal modules over the Lie conformal algebras $\mathcal{W}(b)$.

math.RA↗

arXiv2017

Classification of finite irreducible conformal modules over a class of Lie conformal algebras of Block type

We classify finite irreducible conformal modules over a class of infinite Lie conformal algebras ${\frak {B}}(p)$ of Block type, where $p$ is a nonzero complex number. In particular, we obtain that a finite irreducible conformal module over ${\frak {B}}(p)$ may be a nontrivial extension of a finite conformal module over ${\frak {Vir}}$ if $p=-1$, where ${\frak {Vir}}$ is a Virasoro conformal subalgebra of ${\frak {B}}(p)$. As a byproduct, we also obtain the classification of finite irreducible conformal modules over a series of finite Lie conformal algebras ${\frak b}(n)$ for $n\ge1$.

math.RA↗

arXiv2017

Deformations and generalized derivations of Hom-Lie conformal algebras

The purpose of this paper is to extend the cohomology and conformal derivation theories of the classical Lie conformal algebras to Hom-Lie conformal algebras. In this paper, we develop cohomology theory of Hom-Lie conformal algebras and discuss some applications to the study of deformations of regular Hom-Lie conformal algebras. Also, we introduce $α^k$-derivations of multiplicative Hom-Lie conformal algebras and study their properties.

math.RA↗