arXiv subjects

Jiefeng Liu

Publications and source records attributed to Jiefeng Liu.

At least 19 recordsLinked to original sources

arXiv2026

Affine Rota-Baxter groups and affine skew braces

Rota-Baxter groups and skew braces are closely related algebraic structures, both providing set-theoretical solutions to the Yang-Baxter equation. In this paper, we extend these structures to the setting of affine schemes. First, we introduce affine Rota-Baxter groups and, by leveraging the duality between affine groups and Hopf algebras via their coordinate rings, prove the equivalence between affine Rota-Baxter groups and co-Rota-Baxter Hopf algebras. Next, we show affine Rota-Baxter groups can naturally give rise to the affine skew braces defined by Angiono, Galindo, and Vendramin. Conversely, any affine skew brace can be embedded into an affine Rota-Baxter group. By linking these to the relationship between affine skew braces and Hopf co-braces, we give new connections between co-Rota-Baxter Hopf algebras and Hopf co-braces. Finally, we propose the study of solutions to the Yang-Baxter equation within the framework of affine schemes, demonstrating that affine skew braces naturally give rise to such solutions.

math.GR↗

arXiv2026

On the structure of finite Novikov conformal algebras

We classify finite simple Novikov conformal algebras over an algebraically closed field of characteristic zero. The classification consists of the current conformal algebra over the base field and a one-parameter list of Virasoro-like conformal algebras. A semisimple finite Novikov conformal algebra is proved to be a direct sum of simple ones. We also describe deformations and central extensions of finite simple Novikov conformal algebras.

math.RA↗

arXiv2026

Pre-vertex algebras, pre-Poisson vertex algebras and their deformation quantizations

In this paper, we introduce pre-vertex algebras and pre-Poisson vertex algebras as preLie-algebras in the chiral and classical pseudo-tensor categories, respectively. We show that a pre-vertex algebra gives rise to a vertex algebra, that every Rota-Baxter operator on a vertex algebra induces a pre-vertex algebra, and that every pre-vertex algebra can be embedded into a vertex algebra equipped with a Rota-Baxter operator. Moreover, we prove that pre-vertex algebras are equivalent to dendriform vertex algebras. In the Poisson setting, we demonstrate that pre-Poisson vertex algebras are obtained from Rota-Baxter operators on Poisson vertex algebras, from filtrations of pre-vertex algebras, and as classical limits of pre-vertex formal deformations of differential Zinbiel algebras. This extends the classical relationships among Rota-Baxter operators, pre-Lie algebras, and Poisson algebras to the framework of vertex algebras.

math.QA↗

arXiv2026

Hyper relative differential operators on Lie algebras

In this paper, we first introduce the notion of a hyper relative differential operator on a Lie algebra, in which Nijenhuis operators are used to characterize the relative differential operators and their inverse. We then introduce the notions of DN-structures, KN-structures, and KD-structures on Lie algebras and study the relationships between DN-structures, KD-structures, KN-structures, and hyper relative differential operators. Finally, we investigate hyper symplectic structures and hyper Hessian structures from the view point of hyper relative differential operators, and provide equivalent descriptions for both hyper symplectic structures and hyper Hessian structures.

math.RA↗

arXiv2025

Cohomology of Lie conformal algebroids

We study Lie conformal algebroids (LCAd) and their representations using the language of lambda-brackets and Lie conformal algebras. We describe several general constructions, such as the LCAd of conformal derivations CDer(A) of a differential algebra A, the gauge LCAd G(A,M) associated to a differential algebra A and its module M, the current LCAd F^ of a Lie algebroid F, and the LCAd structure of the space Omega(V) of Kahler differentials over a Poisson vertex algebra (PVA) V. We develop the cohomology theories of LCAd and we relate them to the corresponding cohomology theories of PVA. In particular, we find an isomorphism between the cohomology of a PVA V with coefficients in a module M and the corresponding cohomology of the LCAd Omega(V) with coefficients in the same module.

math.RT↗

arXiv2025

Cohomology of Restricted Poisson algebras in characteristic 2

In this paper, we study restricted Poisson algebras in characteristic 2 and their relationship with restricted Lie-Rinehart algebras, for which we develop a cohomology theory and investigate abelian extensions. We also construct a full cohomology complex for restricted Poisson algebras in characteristic 2 that captures formal deformations and prove that it is isomorphic to the cohomology complex of a suitable restricted Lie-Rinehart algebra, under certain assumptions. A number of examples are provided in order to illustrate our constructions.

math.RT↗

arXiv2025

Pre-Lie 2-bialgebras and 2-grade classical Yang-Baxter equations

We introduce a notion of a para-K\"{a}hler strict Lie 2-algebra, which can be viewed as a categorification of a para-K\"{a}hler Lie algebra. In order to study para-K\"{a}hler strict Lie 2-algebra in terms of strict pre-Lie 2-algebras, we introduce the Manin triples, matched pairs and bialgebra theory for strict pre-Lie 2-algebras and the equivalent relationships between them are also established. By means of the cohomology theory of Lie 2-algebras, we study the coboundary strict pre-Lie 2-algebras and introduce 2-graded classical Yang-Baxter equations in strict pre-Lie 2-algebras. The solutions of the 2-graded classical Yang-Baxter equations are useful to construct strict pre-Lie 2-algebras and para-K\"{a}hler strict Lie 2-algebras. In particular, there is a natural construction of strict pre-Lie 2-bialgebras from the strict pre-Lie 2-algebras.

math.QA↗

arXiv2024

From pre-Lie bialgebras to phase spaces of Lie algebras: a categorical correspondence

This paper establishes a categorical framework for phase spaces of Lie algebras, pre-Lie bialgebras, Manin triples, classical s-matrices, and relative Rota-Baxter operators by introducing the concept of coherent homomorphisms. Starting with endo pre-Lie algebras (pre-Lie algebras equipped with endomorphisms), we extend classical constructions to this enhanced setting, which leads to the notion of coherent endomorphisms for each class of structures. Through polarization, these endomorphisms naturally generalize to coherent homomorphisms, establishing well-defined categories of these algebraic objects. Furthermore, mappings between categories are elevated to functors or equivalences, formalizing interconnections among the constructions. Finally, exploiting the categorical correspondence between s-matrices and relative Rota-Baxter operators, we develop cohomology and deformation of s-matrices, thereby bridging algebraic structures with category-theoretic methods.

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

Cohomology and deformation quantization of Poisson conformal algebras

In this paper, we first recall the notion of (noncommutative) Poisson conformal algebras and describe some constructions of them. Then we study the formal distribution (noncommutative) Poisson algebras and coefficient (noncommutative) Poisson algebras. Next, we introduce the notion of conformal formal deformations of commutative associative conformal algebras and show that Poisson conformal algebras are the corresponding semi-classical limits. At last, we develop the cohomology theory of noncommutative Poisson conformal algebras and use this cohomology to study their deformations.

math.QA↗

arXiv2022

Deformations and Cohomologies of Relative Rota-Baxter Operators on Lie Algebroids and Koszul-Vinberg Structures

Given a Lie algebroid with a representation, we construct a graded Lie algebra whose Maurer-Cartan elements characterize relative Rota-Baxter operators on Lie algebroids. We give the cohomology of relative Rota-Baxter operators and study infinitesimal deformations and extendability of order $n$ deformations to order $n+1$ deformations of relative Rota-Baxter operators in terms of this cohomology theory. We also construct a graded Lie algebra on the space of multi-derivations of a vector bundle whose Maurer-Cartan elements characterize left-symmetric algebroids. We show that there is a homomorphism from the controlling graded Lie algebra of relative Rota-Baxter operators on Lie algebroids to the controlling graded Lie algebra of left-symmetric algebroids. Consequently, there is a natural homomorphism from the cohomology groups of a relative Rota-Baxter operator to the deformation cohomology groups of the associated left-symmetric algebroid. As applications, we give the controlling graded Lie algebra and the cohomology theory of Koszul-Vinberg structures on left-symmetric algebroids.

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

$F$-algebroids and deformation quantization via pre-Lie algebroids

In this paper, first we introduce a new approach to the notion of $F$-algebroids, which is a generalization of $F$-manifold algebras and $F$-manifolds, and show that $F$-algebroids are the corresponding semi-classical limits of pre-Lie formal deformations of commutative associative algebroids. Then we use the deformation cohomology of pre-Lie algebroids to study pre-Lie infinitesimal deformations and extension of pre-Lie $n$-deformations to pre-Lie $(n+1)$-deformations of a commutative associative algebroid. Next we develop the theory of Dubrovin's dualities of $F$-algebroids with eventual identities and use Nijenhuis operators on $F$-algebroids to construct new $F$-algebroids. Finally we introduce the notion of pre-$F$-algebroids, which is a generalization of $F$-manifolds with compatible flat connections. Dubrovin's dualities of pre-$F$-algebroids with eventual identities, Nijenhuis operators on pre-$F$-algebroids and their applications to integral systems are discussed.

math-ph↗

arXiv2022

Rota-Baxter Lie $2$-algebras

In this paper, we introduce the notion of Rota-Baxter Lie $2$-algebras, which is a categorification of Rota-Baxter Lie algebras. We prove that the category of Rota-Baxter Lie $2$-algebras and the category of $2$-term Rota-Baxter $L_\infty$-algebras are equivalent. We introduce the notion of a crossed module of Rota-Baxter Lie algebras and show that there is a one-to-one correspondence between strict $2$-term Rota-Baxter $L_\infty$-algebras and crossed modules of Rota-Baxter Lie algebras. We give the construction of crossed modules of Lie algebras from crossed modules of Rota-Baxter Lie algebras.

math.CT↗

arXiv2021

Cohomologies and deformations of pre-Lie-morphism triples

A (pre-)Lie-morphism triple consists of two (pre-)Lie algebras and a (pre-)Lie algebra homomorphism between them. We give chomologies of pre-Lie-morphism triples. As an application, we study the infinitesimal deformations of pre-Lie-morphism triples. Finally, we show that the cohomology of the pre-Lie-morphism triple can be deduced from a new cohomology of the Lie-morphism triple.

math.RA↗

arXiv2021

Cohomologies and crossed modules for pre-Lie Rinehart algebras

A pre-Lie-Rinehart algebra is an algebraic generalization of the notion of a left-symmetric algebroid. We construct pre-Lie-Rinehart algebras from r-matrices through Lie algebra actions. We study cohomologies of pre-Lie-Rinehart algebras and show that abelian extensions of pre-Lie-Rinehart algebras are classified by the second cohomology groups. We introduce the notion of crossed modules for pre-Lie-Rinehart algebras and show that they are classified by the third cohomology groups of pre-Lie-Rinehart algebras. At last, we use (pre-)Lie-Rinehart 2-algebras to characterize the crossed modules for (pre-)Lie Rinehart algebras.

math.RA↗

arXiv2021

Cohomologies of 3-Lie algebras with derivations

In this paper, we consider a 3-Lie algebra with a derivation (called a 3-LieDer pair). We define cohomology for a 3-LieDer pair with coefficients in a representation. We use this cohomology to study deformations and abelian extensions of 3-LieDer pairs. We give the notion of a 3-Lie2Der pair, which can be viewed as the categorification of a 3-LieDer pair. We show that skeletal 3-Lie2Der pairs are classified by triples given by 3-LieDer pairs, representations and 3-cocycles. We define crossed modules of 3-LieDer pairs and show that there exists a one-to-one correspondence between strict 3-Lie2Der pairs and crossed modules of 3-LieDer pairs.

math.RA↗

arXiv2021

Admissible Poisson bialgebras

An admissible Poisson algebra (or briefly, an adm-Poisson algebra) gives an equivalent presentation with only one operation for a Poisson algebra. We establish a bialgebra theory for adm-Poisson algebras independently and systematically, including but beyond the corresponding results on Poisson bialgebras given in [27]. Explicitly, we introduce the notion of adm-Poisson bialgebras which are equivalent to Manin triples of adm-Poisson algebras as well as Poisson bialgebras. The direct correspondence between adm-Poisson bialgebras with one comultiplication and Poisson bialgebras with one cocommutative and one anti-cocommutative comultiplications generalizes and illustrates the polarization-depolarization process in the context of bialgebras. The study of a special class of adm-Poisson bialgebras which include the known coboundary Poisson bialgebras in [27] as a proper subclass in general, illustrating an advantage in terms of the presentation with one operation, leads to the introduction of adm-Poisson Yang-Baxter equation in an adm-Poisson algebra. It is an unexpected consequence that both the adm-Poisson Yang-Baxter equation and the associative Yang-Baxter equation have the same form and thus it motivates and simplifies the involved study from the study of the associative Yang-Baxter equation, which is another advantage in terms of the presentation with one operation. A skew-symmetric solution of adm-Poisson Yang-Baxter equation gives an adm-Poisson bialgebra. Finally the notions of an $\mathcal O$-operator of an adm-Poisson algebra and a pre-adm-Poisson algebra are introduced to construct skew-symmetric solutions of adm-Poisson Yang-Baxter equation and hence adm-Poisson bialgebras. Note that a pre-adm-Poisson algebra gives an equivalent presentation for a pre-Poisson algebra introduced by Aguiar.

math.QA↗