arXiv subjects

Yunhe Sheng

Publications and source records attributed to Yunhe Sheng.

At least 19 recordsLinked to original sources

arXiv2026

Cohomology theory of Novikov algebras and applications

In this paper, first we give a new characterization of the cohomology of pre-Lie algebras using the Chevalley-Eilenberg cohomology associated to a morphism from the operad of Lie algebras to Hadamard product of the operad of pre-Lie algebras and its Koszul dual operad. Then we apply the same approach to study the cohomology of Novikov algebras, and give the cochain complex explicitly. The cochain complex of the underlying pre-Lie algebra is shown to be isomorphic to the quotient of the cochain complex of a Novikov algebra. Consequently, there is a long exact sequence connecting the cohomologies of a Novikov algebra and the underlying pre-Lie algebra. The cohomology of a Novikov algebra with coefficients in a representation is introduced using pseudo-tensor categories. As applications, we show that infinitesimal deformations and abelian extensions are classified by the second cohomology groups with different coefficients. Various examples are given to illustrate the difference between the cohomology of a Novikov algebra and that of the underlying pre-Lie algebra.

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arXiv2026

From Lie--Rinehart Algebras to $F$-Manifold Algebras

For every Lie--Rinehart algebra, we construct an $F$-manifold algebra on the direct sum of its base algebra and module. Contrary to the assertion in \cite[Proposition 13.3.26]{LodayVallette}, the resulting structure is generally not Poisson. We determine when powers of the positive-degree ideal in the associated symmetric Poisson algebra are Poisson ideals, and relate the Leibnizator to the Lie--Rinehart differential. For a finite projective module of constant rank, the trace of the Leibnizator recovers the anchor and yields a rigidity result for injective anchors. We conclude with algebraic and geometric examples.

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arXiv2026

Equivariant Nijenhuis Lie Algebras: Extensions to Classical Lie-Theoretic Structures

We develop a theory of equivariant Nijenhuis Lie algebras (ENL algebras), namely Lie algebras equipped with Nijenhuis operators satisfying an equivariance condition with respect to the adjoint representation. This compatibility condition allows classical Lie bialgebra constructions to extend naturally to the operator-equipped setting. Within this framework, we define ENL bialgebras and establish the associated notions of matched pairs, Manin triples, and Drinfel'd doubles. We show that coboundary ENL bialgebras are characterized by EN $r$-matrices satisfying an equivariant classical Yang-Baxter equation. We further introduce EN-relative Rota-Baxter operators and prove that they provide an operator-theoretic realization of such $r$-matrices, leading to descendant ENL algebras and to solutions of the classical Yang-Baxter equation on semidirect ENL algebras. In the quadratic case, this construction recovers Rota-Baxter operators of weight zero. Finally, we extend the EN framework to pre-Lie algebras and show that pre-ENL algebras naturally induce associated ENL structures.

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arXiv2025

Post-Hopf algebroids, post-Lie-Rinehart algebras and geometric numerical integration

In this paper, we introduce the notion of post-Hopf algebroids, generalizing the pre-Hopf algebroids introduced in [Bronasco, Laurent, 2025] in the study of exotic aromatic S-series. We construct action post-Hopf algebroids through actions of post-Hopf algebras. We show that the universal enveloping algebra of a post-Lie-Rinehart algebra (post-Lie algebroid) is naturally a post-Hopf algebroid. As a byproduct, we construct the free post-Lie-Rinehart algebra using a magma algebra with a linear map to the derivation Lie algebra of a commutative associative algebra. Applications in geometric numerical integration on manifolds are given.

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arXiv2025

Rota-Baxter operators on braces, post-braces and the Yang-Baxter equation

Combining the notions of braces and relative Rota-Baxter operators on groups in connection with the Yang-Baxter equation and a factorization theorem of Lie groups from integrable systems, relative Rota-Baxter operators on braces and post-braces are introduced. A relative Rota-Baxter operator on a brace naturally induces a post-brace, and conversely, every post-brace determines a relative Rota-Baxter operator on its sub-adjacent brace. Furthermore, a post-brace yields two Drinfel'd-isomorphic solutions to the Yang-Baxter equation. As a special case, {\it enhanced} relative Rota-Baxter operators give rise to matched pairs of braces. Focusing on enhanced Rota-Baxter operators on two-sided braces, a corresponding factorization theorem is established. Examples are provided from the two-sided brace associated with the three-dimensional Heisenberg Lie algebra.

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arXiv2025

Braided dynamical groups, the dynamical Yang-Baxter equation and related structures

We introduce the notion of a braided dynamical group which is a matched pair of dynamical groups satisfying extra conditions. It is shown to give a solution of the dynamical Yang-Baxter equation and at the same time a braided groupoid, thereby integrating the approaches of Andruskiewitsch and Matsumoto-Shimizu respectively that use these two notions to produce quiver-theoretical solutions of the Yang-Baxter equation. We pursue this connection further by relative Rota-Baxter operators on dynamical groups, which give rise to matched pairs of dynamical groups. As the derived structures of relative Rota-Baxter operators on dynamical groups, dynamical post-groups are introduced and are shown to be equivalent to braided dynamical groups. Finally, skew-braces are generalized to dynamical skew-braces as another equivalent notion of braided dynamical groups.

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arXiv2025

Reflections on Rota-Baxter Lie algebras, the classical reflection equation and Poisson homogeneous spaces

In this paper, first we introduce the notion of reflections on quadratic Rota-Baxter Lie algebras of weight $\lambda$, and show that they give rise to solutions of the classical reflection equation for the corresponding triangular Lie bialgebra ($\lambda=0$) and factorizable Lie bialgebra ($\lambda\neq0$). Then we study reflections on relative Rota-Baxter Lie algebras, and also show that they give rise to solutions of the classical reflection equation for certain Lie bialgebras determined by the relative Rota-Baxter operators. In particular, involutive automorphisms on pre-Lie algebras and post-Lie algebras naturally lead to reflections on the induced relative Rota-Baxter Lie algebras. Finally, we derive Poisson Lie groups and Poisson homogeneous spaces from quadratic Rota-Baxter Lie algebras and relative Rota-Baxter Lie algebras.

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arXiv2025

Leibniz $2$-algebras, linear $2$-racks and the Zamolodchikov Tetrahedron equation

In this paper, first we show that a central Leibniz 2-algebra naturally gives rise to a solution of the Zamolodchikov Tetrahedron equation. Then we introduce the notion of linear 2-racks and show that a linear 2-rack also gives rise to a solution of the Zamolodchikov Tetrahedron equation. We show that a central Leibniz 2-algebra gives rise to a linear 2-rack if the underlying 2-vector space is splittable. Finally we discuss the relation between linear 2-racks and 2-racks, and show that a linear 2-rack gives rise to a 2-rack structure on the group-like category. A concrete example of strict 2-racks is constructed from an action of a strict 2-group.

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arXiv2025

Post-Lie deformations of pre-Lie algebras and their applications in Regularity Structures

In this paper, we study post-Lie deformations of a pre-Lie algebra, namely deforming a pre-Lie algebra into a post-Lie algebra. We construct the differential graded Lie algebra that governs post-Lie deformations of a pre-Lie algebra. We also develop the post-Lie cohomology theory for a pre-Lie algebra, by which we classify infinitesimal post-Lie deformations of a pre-Lie algebra using the second cohomology group. The rigidity of such kind of deformations is also characterized using the second cohomology group. Finally, we apply this deformation theory to Regularity Structures. We prove that the post-Lie algebraic structure on the decorated trees which appears spontaneously in Regularity Structures is a post-Lie deformation of a pre-Lie algebra.

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arXiv2025

Phase space of a Poisson algebra and the induced Pre-Poisson bialgebra

In this paper, we first introduce the notion of a phase space of a Poisson algebra, and show that a Poisson algebra has a phase space if and only if it is sub-adjacent to a pre-Poisson algebra. Moreover, we introduce the notion of Manin triples of pre-Poisson algebras and show that there is a one-to-one correspondence between Manin triples of pre-Poisson algebras and phase spaces of Poisson algebras. Then we introduce the notion of pre-Poisson bialgebras, which is equivalent to Manin triples of pre-Poisson algebras. We study coboundary pre-Poisson bialgebras, which leads to an analogue of the classical Yang-Baxter equation. Furthermore, we introduce the notions of quasi-triangular and factorizable pre-Poisson bialgebras as special cases. A quasi-triangular pre-Poisson bialgebra gives rise to a relative Rota-Baxter operator of weight $1$. The double of a pre-Poisson bialgebra enjoys a natural factorizable pre-Poisson bialgebra structure. Finally, we introduce the notion of quadratic Rota-Baxter pre-Poisson algebras and show that there is a one-to-one correspondence between quadratic Rota-Baxter pre-Poisson algebras and factorizable pre-Poisson bialgebras. Based on this construction, we give a phase space for a Rota-Baxter symplectic Poisson algebra.

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arXiv2025

Homotopy theory of post-Lie algebras

In this paper, we study the homotopy theory of post-Lie algebras. Guided by Koszul duality theory, we consider the graded Lie algebra of coderivations of the cofree conilpotent graded cocommutative cotrialgebra generated by $V$. We show that in the case of $V$ being a shift of an ungraded vector space $W$, Maurer-Cartan elements of this graded Lie algebra are exactly post-Lie algebra structures on $W$. The cohomology of a post-Lie algebra is then defined using Maurer-Cartan twisting. The second cohomology group of a post-Lie algebra has a familiar interpretation as equivalence classes of infinitesimal deformations. Next we define a post-Lie$_\infty$ algebra structure on a graded vector space to be a Maurer-Cartan element of the aforementioned graded Lie algebra. Post-Lie$_\infty$ algebras admit a useful characterization in terms of $L_\infty$-actions (or open-closed homotopy Lie algebras). Finally, we introduce the notion of homotopy Rota-Baxter operators on open-closed homotopy Lie algebras and show that certain homotopy Rota-Baxter operators induce post-Lie$_\infty$ algebras.

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arXiv2024

Cohomology of Lie coalgebras

A Koszul duality-type correspondence between coderived categories of conilpotent differential graded Lie coalgebras and their Chevalley-Eilenberg differential graded algebras is established. This gives an interpretation of Lie coalgebra cohomology as a certain kind of derived functor. A similar correspondence is proved for coderived categories of commutative cofibrant differential graded algebras and their Harrison differential graded Lie coalgebras.

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arXiv2024

On the integration of relative Rota-Baxter Lie algebras

In this paper, we give the necessary and sufficient conditions of the integrability of relative Rota-Baxter Lie algebras via double Lie groups, matched pairs of Lie groups and factorization of diffeomorphisms respectively. We use the integrability of Rota-Baxter operators to characterize whether the Poisson-Lie group integrating a factorizable Lie bialgebra is again factorizable. We thoroughly study the integrability of Rota-Baxter operators on the unique nontrivial 2-dimensional Lie algebra. As a byproduct, we construct a matched pair of Lie algebras that can not be integrated to a matched pair of Lie groups.

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arXiv2024

The Yang-Baxter equation, Leibniz algebras, racks and related algebraic structures

The purpose of this paper is to clarify the relations between various constructions of solutions of the Yang-Baxter equation from Leibniz algebras, racks, 3-Leibniz algebras, 3-racks, linear racks, trilinear racks, and give new constructions of solutions of the Yang-Baxter equation. First we show that a 3-Leibniz algebra naturally gives rise to a 3-rack on the underlying vector space, which generalizes Kinyon's construction of racks from Leibniz algebras. Then we show that a trilinear rack naturally gives rise to a linear rack. Combined with Lebed's construction of solutions of the Yang-Baxter equation from linear racks, our results give an intrinsic explanation of Abramov and Zappala's construction of solutions of the Yang-Baxter equation from trilinear racks. Next we show that a 3-Leibniz algebra gives rise to a trilinear rack, which generalizes Abramov and Zappala's construction from 3-Lie algebras. Finally, we construct solutions of the Yang-Baxter equation using central extensions of 3-Leibniz algebras and Leibniz algebras. In particular, given a 3-Leibniz algebra, there are two different approaches to construct solutions of the Yang-Baxter equation, namely either consider the central extension of the Leibniz algebra on the fundamental objects, or consider the Leibniz algebra on the fundamental objects of the central extension of the 3-Leibniz algebra. We also show that there is a homomorphism between the corresponding solutions.

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arXiv2024

Post-groupoids and quiver-theoretical solutions of the Yang-Baxter equation

The notion of post-groups was introduced by Bai, Guo and the first two authors recently, which are the global objects corresponding to post-Lie algebras, equivalent to skew-left braces, and can be used to construct set-theoretical solutions of the Yang-Baxter equation. In this paper, first we introduce the notion of post-groupoids, which consists of a group bundle and some other structures satisfying some compatibility conditions. Post-groupoids reduce to post-groups if the underlying base is one point. An action of a group on a set gives rise to the natural example of post-groupoids. We show that a post-groupoid gives rise to a groupoid (called the Grossman-Larson groupoid), and an action on the original group bundle. Then we introduce the notion of relative Rota-Baxter operators on a groupoid with respect to an action on a group bundle. A relative Rota-Baxter operator naturally gives rise to a post-groupoid and a matched pair of groupoids. One important application of post-groupoids is that they give rise to quiver-theoretical solutions of the Yang-Baxter equation on the quiver underlying the Grossman-Larson groupoid. We also introduce the notion of a skew-left bracoid, which consists of a group bundle and a groupoid satisfying some compatibility conditions. A skew-left bracoid reduces to a skew-left brace if the underlying base is one point. We give the one-to-one correspondence between post-groupoids and skew-left bracoids. Finally, we show that post-Lie groupoids give rise to post-Lie algebroids via differentiation.

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arXiv2024

Quasi-triangular, factorizable Leibniz bialgebras and relative Rota-Baxter operators

We introduce the notion of quasi-triangular Leibniz bialgebras, which can be constructed from solutions of the classical Leibniz Yang-Baxter equation (CLYBE) whose skew-symmetric parts are invariant. In addition to triangular Leibniz bialgebras, quasi-triangular Leibniz bialgebras contain factorizable Leibniz bialgebras as another subclass, which lead to a factorization of the underlying Leibniz algebras. Relative Rota-Baxter operators with weights on Leibniz algebras are used to characterize solutions of the CLYBE whose skew-symmetric parts are invariant. On skew-symmetric quadratic Leibniz algebras, such operators correspond to Rota-Baxter type operators. Consequently, we introduce the notion of skew-symmetric quadratic Rota-Baxter Leibniz algebras, such that they give rise to triangular Leibniz bialgebras in the case of weight $0$, while they are in one-to-one correspondence with factorizable Leibniz bialgebras in the case of nonzero weights.

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arXiv2024

Stability and rigidity of 3-Lie algebra morphisms

In this paper, first we use the higher derived brackets to construct an $L_\infty$-algebra, whose Maurer-Cartan elements are $3$-Lie algebra morphisms. Using the differential in the $L_\infty$-algebra that govern deformations of the morphism, we give the cohomology of a $3$-Lie algebra morphism. Then we study the rigidity and stability of $3$-Lie algebra morphisms using the established cohomology theory. In particular, we show that if the first cohomology group is trivial, then the morphism is rigid; if the second cohomology group is trivial, then the morphism is stable. Finally, we study the stability of $3$-Lie subalgebras similarly.

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arXiv2024

Deformation maps of quasi-twilled Lie algebras

In this paper, we provide a unified approach to study the cohomology theories and deformation theories of various types of operators in the category of Lie algebras, including modified $r$-matrices, crossed homomorphisms, derivations, homomorphisms, relative Rota-Baxter operators, twisted Rota-Baxter operators, Reynolds operators and deformation maps of matched pairs of Lie algebras. The main ingredients are quasi-twilled Lie algebras. We introduce two types of deformation maps of a quasi-twilled Lie algebra. Deformation maps of type I unify modified $r$-matrices, crossed homomorphisms, derivations and homomorphisms between Lie algebras, while deformation maps of type II unify relative Rota-Baxter operators, twisted Rota-Baxter operators, Reynolds operators and deformation maps of matched pairs of Lie algebras. We further give the controlling algebras and cohomologies of these two types of deformation maps, which not only recover the existing results for crossed homomorphisms, derivations, homomorphisms, relative Rota-Baxter operators, twisted Rota-Baxter operators and Reynolds operators, but also leads to some new results which are unable to solve before, e.g. the controlling algebras and cohomologies of modified $r$-matrices and deformation maps of matched pairs of Lie algebras.

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