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Yanyong Hong

Publications and source records attributed to Yanyong Hong.

At least 19 recordsLinked to original sources
arXiv2026

On $q$-pre-Lie algebras

In this paper, we introduce the notion of $q$-pre-Lie algebras from the perspective of representations of Lie algebras, providing a parametrized generalization that unifies pre-Lie algebras and anti-pre-Lie algebras. For a $q$-pre-Lie algebra $(A,\circ)$, the commutator of $\circ$ is a Lie bracket and the left multiplication operator scaled by $q$ gives a representation of the associated commutator Lie algebra. We also introduce the notions of $q$-$\mathcal{O}$-operators and $q$-Novikov algebras, and investigate their relationships with $q$-pre-Lie algebras. Several explicit constructions of $q$-pre-Lie algebras are provided. Moreover, we give a complete classification of graded $q$-pre-Lie algebra structures on the Witt algebra and prove the existence of such structures on the Virasoro algebra only when $q=1$ and $q=2$. Finally, we classify compatible root-graded $q$-pre-Lie algebra structures on finite-dimensional complex simple Lie algebras.

math.RA↗
arXiv2026

Solvability and nilpotency of transposed Novikov-Poisson algebras

In this paper, we develop the theory of nilpotency and solvability for transposed Novikov-Poisson algebras. We first establish several equivalent conditions for a dialgebra to be nilpotent, and show that the lower central series of a transposed Novikov-Poisson algebra $P$ admits a simplified form. We then prove that $P$ is solvable if and only if it is right nilpotent, and also if and only if $P^2$ is nilpotent. Moreover, we show that nilpotency (respectively, solvability) of a transposed Novikov-Poisson algebra is equivalent to nilpotency (respectively, solvability) of both its underlying commutative associative algebra and its underlying Novikov algebra. Finally, we prove that It\^{o}'s theorem holds for transposed Novikov-Poisson algebras.

math.RA↗
arXiv2026

Nilpotency and Frattini theory for transposed Poisson algebras

We develop the theory of nilpotency and the Frattini theory for transposed Poisson algebras. The lower central series is shown to admit a simplified form, and an analogue of Engel's theorem is established: a finite-dimensional transposed Poisson algebra is nilpotent precisely when the left multiplication operators in both the associative and the Lie structures are nilpotent. Constructions of nilpotent and solvable algebras via tensor products and derivations are given. For a finite-dimensional Lie-nilpotent transposed Poisson algebra, we prove that the derived Lie subalgebra is a nilpotent ideal, which implies that the nilpotent radical coincides with the associative radical. In the framework of Frattini theory, we show that the Frattini subalgebra is always contained in the derived algebra and the Frattini ideal is associative nilpotent. When the algebra is nilpotent, all maximal subalgebras are ideals and the Frattini subalgebra equals the derived algebra. Conversely, for a Lie-nilpotent transposed Poisson algebra, if all maximal subalgebras are ideals, the algebra either is nilpotent or decomposes as a direct sum of a one-dimensional algebra generated by an idempotent and the nilpotent radical; if the Frattini subalgebra equals the derived algebra, the algebra is necessarily nilpotent. We also prove that the zero socle coincides with the nilpotent radical, and when the Frattini ideal is zero, the algebra splits into a subalgebra and its zero socle; in the Lie-nilpotent case this subalgebra is abelian as a Lie algebra.

math.RA↗
arXiv2026

Jacobi algebras and Jacobi Novikov-Poisson algebras

In this paper, we introduce the notion of Jacobi Novikov-Poisson algebras and demonstrate that their affinization yields Jacobi algebras. We note that every unital differential Novikov-Poisson algebra is also a Jacobi Novikov-Poisson algebra. Additionally, any Jacobi Novikov-Poisson algebra gives rise to a Jacobi algebra, either by taking the commutator bracket of its underlying Novikov algebra or by using a derivation. We provide classifications of low-dimensional Jacobi Novikov-Poisson algebras including those of dimensions 2 and 3 over $\mathbb{C}$ up to isomorphism and show that the tensor product of two such algebras remains a Jacobi Novikov-Poisson algebra. Several further constructions of Jacobi Novikov-Poisson algebras from existing ones are also presented. The notion of Frobenius Jacobi Novikov-Poisson algebras is introduced, and several equivalent characterizations are established in terms of quadratic structures and integrals. Classifications of quadratic Jacobi Novikov-Poisson algebras of dimensions 2 and 3 over $\mathbb{C}$ are given. Finally, we provide an explicit construction of Frobenius Jacobi algebras using finite-dimensional quadratic Jacobi Novikov-Poisson algebras and finite-dimensional quadratic right Jacobi Novikov-Poisson algebras.

math.RA↗
arXiv2026

Transposed Novikov-Poisson algebras

In this paper, we introduce the definition of transposed Novikov-Poisson algebras, whose affinization are transposed Poisson algebras. Moreover, we show that there is a natural transposed Poisson algebra structure on the tensor product of a transposed Novikov-Poisson algebra and a right differential Novikov-Poisson algebra. A transposed Poisson algebra also naturally arises from a transposed Novikov-Poisson algebra by taking the commutator Lie algebra of the Novikov algebra. We show that the tensor products of two transposed Novikov-Poisson algebras are also transposed Novikov-Poisson algebras. Several constructions of transposed Novikov-Poisson algebras are presented. Moreover, transposed Novikov-Poisson algebras are closely related to $\frac{1}{2}$-derivations of the associated Novikov algebras. By using $\frac{1}{2}$-derivations, we show that there are non-trivial transposed Novikov-Poisson algebra structures on solvable Novikov algebras with some conditions. We also prove that if a non-trivial transposed Novikov-Poisson algebra is simple, then the associated Novikov algebra is simple. Therefore, if the base field is algebraically closed and of characteristic 0, then any simple transposed Novikov-Poisson is of dimension $1$. Transposed Novikov-Poisson algebra structures on some simple Novikov algebras are also characterized.

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arXiv2025

Infinite-dimensional pre-Lie bialgebras via affinization of pre-Novikov bialgebras

In this paper, we show that there is a pre-Lie algebra structure on the tensor product of a pre-Novikov algebra and a right Novikov dialgebra and the tensor product of a pre-Novikov algebra and a special right Novikov algebra on the vector space of Laurent polynomials being a pre-Lie algebra characterizes the pre-Novikov algebra. The latter is called the affinization of a pre-Novikov algebra. Moreover, we extend this construction of pre-Lie algebras and the affinization of pre-Novikov algebras to the context of bialgebras. We show that there is a completed pre-Lie bialgebra structure on the tensor product of a pre-Novikov bialgebra and a quadratic Z-graded right Novikov algebra. Moreover, a pre-Novikov bialgebra can be characterized by the fact that its affinization by a quadratic Z-graded right Novikov algebra on the vector space of Laurent polynomials gives an infinite-dimensional completed pre-Lie bialgebras. Note that the reason why we choose a quadratic right Novikov algebra instead of a right Novikov dialgebra with a special bilinear form is also given. Furthermore, we construct symmetric completed solutions of the S-equation in the induced pre-Lie algebra by symmetric solutions of the pre-Novikov Yang-Baxter equation in a pre-Novikov algebra.

math.RA↗
arXiv2025

Extended $\mathcal{O}$-operators, Novikov Yang-Baxter equations and post-Novikov algebras

In this paper, we introduce the definition of extended $\mathcal{O}$-operators on a Novikov algebra $(A,\circ)$ associated to an $A$-bimodule Novikov algebra which is a generalization of the definition of $\mathcal{O}$-operators and show that there are new Novikov algebra structures on the $A$-bimodule Novikov algebra obtained from extended $\mathcal{O}$-operators. We also introduce the definition of post-Novikov algebras and show that there is a close relationship between post-Novikov algebras and $\mathcal{O}$-operators of weight $\lambda$. The tensor form of extended $\mathcal{O}$-operators is also investigated which leads to the definition of extended Novikov Yang-Baxter equations, which is a generalization of the notion of Novikov Yang-Baxter equations. The relationships between extended $\mathcal{O}$-operators, Novikov Yang-Baxter equations, extended Novikov Yang-Baxter equations and generalized Novikov Yang-Baxter equations are established.

math.RA↗
arXiv2025

On quadratic Novikov algebras

A quadratic Novikov algebra is a Novikov algebra $(A, \circ)$ with a symmetric and nondegenerate bilinear form $B(\cdot,\cdot)$ satisfying $B(a\circ b, c)=-B(b, a\circ c+c\circ a)$ for all $a$, $b$, $c\in A$. This notion appeared in the theory of Novikov bialgebras. In this paper, we first investigate some properties of quadratic Novikov algebras and give a decomposition theorem of quadratic Novikov algebras. Then we present a classification of quadratic Novikov algebras of dimensions $2$ and $3$ over $\mathbb{C}$ up to isomorphism. Finally, a construction of quadratic Novikov algebras called double extension is presented and we show that any quadratic Novikov algebra containing a nonzero isotropic ideal can be obtained by double extensions. Based on double extension, an example of quadratic Novikov algebras of dimension 4 is given.

math.RA↗
arXiv2025

Leibniz conformal bialgebras and the classical Leibniz conformal Yang-Baxter equation

We introduce the notion of Leibniz conformal bialgebras, presenting a bialgebra theory for Leibniz conformal algebras as well as the conformal analogues of Leibniz bialgebras. They are equivalently characterized in terms of matched pairs and conformal Manin triples of Leibniz conformal algebras. In the coboundary case, the classical Leibniz conformal Yang-Baxter equation is introduced, whose symmetric solutions give Leibniz conformal bialgebras. Moreover, such solutions are constructed from $\mathcal{O}$-operators on Leibniz conformal algebras and Leibniz-dendriform conformal algebras. On the other hand, the notion of Novikov bi-dialgebras is introduced, which correspond to a class of Leibniz conformal bialgebras, lifting the correspondence between Novikov dialgebras and a class of Leibniz conformal algebras to the context of bialgebras. In addition, we introduce the notion of classical duplicate Novikov Yang-Baxter equation whose symmetric solutions produce Novikov bi-dialgebras and thus Leibniz conformal bialgebras.

math.RA↗
arXiv2024

On Poisson conformal bialgebras

We develop a conformal analog of the theory of Poisson bialgebras as well as a bialgebra theory of Poisson conformal algebras. We introduce the notion of Poisson conformal bialgebras, which are characterized by Manin triples of Poisson conformal algebras. A class of special Poisson conformal bialgebras called coboundary Poisson conformal bialgebras are constructed from skew-symmetric solutions of the Poisson conformal Yang-Baxter equation, whose operator forms are studied. Then we show that the semi-classical limits of conformal formal deformations of commutative and cocommutative antisymmetric infinitesimal conformal bialgebras are Poisson conformal bialgebras. Finally, we extend the correspondence between Poisson conformal algebras and Poisson-Gel'fand-Dorfman algebras to the context of bialgebras, that is, we introduce the notion of Poisson-Gel'fand-Dorfman bialgebras and show that Poisson-Gel'fand-Dorfman bialgebras correspond to a class of Poisson conformal bialgebras. Moreover, a construction of Poisson conformal bialgebras from pre-Poisson-Gel'fand-Dorfman algebras is given.

math.RA↗
arXiv2024

Quasi-Frobenius Novikov algebras and pre-Novikov bialgebras

Pre-Novikov algebras and quasi-Frobenius Novikov algebras naturally appear in the theory of Novikov bialgebras. In this paper, we show that there is a natural pre-Novikov algebra structure associated to a quasi-Frobenius Novikov algebra. Then we introduce the definition of double constructions of quasi-Frobenius Novikov algebras associated to two pre-Novikov algebras and show that it is characterized by a pre-Novikov bialgebra. We also introduce the notion of pre-Novikov Yang-Baxter equation, whose symmetric solutions can produce pre-Novikov bialgebras. Moreover, the operator forms of pre-Novikov Yang-Baxter equation are also investigated.

math.RA↗
arXiv2024

Deformation families of Novikov bialgebras via differential antisymmetric infinitesimal bialgebras

Generalizing S. Gelfand's classical construction of a Novikov algebra from a commutative differential algebra, a deformation family $(A,\circ_q)$, for scalars $q$, of Novikov algebras is constructed from what we call an admissible commutative differential algebra, by adding a second linear operator to the commutative differential algebra with certain admissibility condition. The case of $(A,\circ_0)$ recovers the construction of S. Gelfand. This admissibility condition also ensures a bialgebra theory of commutative differential algebras, enriching the antisymmetric infinitesimal bialgebra. This way, a deformation family of Novikov bialgebras is obtained, under the further condition that the two operators are bialgebra derivations. As a special case, we obtain a bialgebra variation of S. Gelfand's construction with an interesting twist: every commutative and cocommutative differential antisymmetric infinitesimal bialgebra gives rise to a Novikov bialgebra whose underlying Novikov algebra is $(A,\circ_{-\frac{1}{2}})$ instead of $(A,\circ_0)$. The close relations of the classical bialgebra theories with Manin triples, classical Yang-Baxter type equations, $\mathcal{O}$-operators, and pre-structures are expanded to the two new bialgebra theories, in a way that is compatible with the just established connection between the two bialgebras. As an application, Novikov bialgebras are obtained from admissible differential Zinbiel algebras.

math.QA↗
arXiv2023

The Heisenberg-Virasoro Lie conformal superalgebra

In this paper, we introduce a finite Lie conformal superalgebra called the Heisenberg-Virasoro Lie conformal superalgebra $\mathfrak{s}$ by using a class of Heisenberg-Virasoro Lie conformal modules. The super Heisenberg-Virasoro algebra of Ramond type $§$ is defined by the formal distribution Lie superalgebra of $\mathfrak{s}$. Then we construct a class of simple $§$-modules, which are induced from simple modules of some finite dimensional solvable Lie superalgebras. These modules are isomorphic to simple restricted $§$-modules, and include the highest weight modules, Whittaker modules and high order Whittaker modules. As a byproduct, we present a subalgebra of $§$, which is isomorphic to the super Heisenberg-Virasoro algebra of Neveu-Schwarz type.

math.RT↗
arXiv2023

Conformal triple derivations and triple homomorphisms of Lie conformal algebras

Let $\mathcal{R}$ be a finite Lie conformal algebra. In this paper, we first investigate the conformal derivation algebra $CDer(\mathcal{R})$, the conformal triple derivation algebra $CTDer(\mathcal{R})$ and the generalized conformal triple derivation algebra $GCTDer(\mathcal{R})$. Mainly, we focus on the connections among these derivation algebras. Next, we give a complete classification of (generalized) conformal triple derivation algebras on all finite simple Lie conformal algebras. In particular, $CTDer(\mathcal{R})= CDer(\mathcal{R})$, where $\mathcal{R}$ is a finite simple Lie conformal algebra. But for $GCDer(\mathcal{R})$, we obtain a conclusion that is closely related to $CDer(\mathcal{R})$. Finally, we introduce the definition of triple homomorphism of a Lie conformal algebra. Furthermore, triple homomorphisms of all finite simple Lie conformal algebras are also characterized.

math.RA↗
arXiv2023

Algebraic constructions for left-symmetric conformal algebras

Let $R$ be a left-symmetric conformal algebra and $Q$ be a $\mathbb{C}[\partial]$-module. We introduce the notion of a unified product for left-symmetric conformal algebras and apply it to construct an object $\mathcal{H}^2_R(Q,R)$ to describe and classify all left-symmetric conformal algebra structures on the direct sum $E=R\oplus Q$ as a $\mathbb{C}[\partial]$-module such that $R$ is a subalgebra of $E$ up to isomorphism whose restriction on $R$ is the identity map. Moreover, we study $\mathcal{H}^2_R(Q,R)$ in detail when $Q$, $R$ are free as $\mathbb{C}[\partial]$-modules and $\text{rank}Q=1$. Some special products such as crossed product and bicrossed product are also investigated.

math.RA↗
arXiv2023

One-dimensional central extensions and simplicities of a class of left-symmetric conformal algebras

In this paper, we introduce the definition of pre-Gel'fand-Dorfman algebra and present several constructions. Moreover, we show that a class of left-symmetric conformal algebras named quadratic left-symmetric conformal algebras are one to one correspondence with pre-Gel'fand-Dorfman algebras. Then we investigate the simplicities and central extensions of quadratic left-symmetric conformal algebras by a one-dimensional centre from the point of view of pre-Gel'fand-Dorfman algebras. We show that under some conditions, central extensions of quadratic left-symmetric conformal algebras by a one-dimensional centre can be characterized by four bilinear forms on pre-Gel'fand-Dorfman algebras. Several methods to construct simple quadratic left-symmetric conformal algebras from pre-Gel'fand-Dorfman algebras are also given.

math.RA↗
arXiv2023

Infinite-dimensional Lie bialgebras via affinization of Novikov bialgebras and Koszul duality

Balinsky and Novikov showed that the affinization of a Novikov algebra naturally defines a Lie algebra, a property that in fact characterizes the Novikov algebra. It is also an instance of the operadic Koszul duality. In this paper, we develop a bialgebra theory for the Novikov algebra, namely the Novikov bialgebra, which is characterized by the fact that its affinization (by a quadratic right Novikov algebra) gives an infinite-dimensional Lie bialgebra, suggesting a Koszul duality for properads. A Novikov bialgebra is also characterized as a Manin triple of Novikov algebras. The notion of Novikov Yang-Baxter equation is introduced, whose skewsymmetric solutions can be used to produce Novikov bialgebras and hence Lie bialgebras. Moreover, these solutions also give rise to skewsymmetric solutions of the classical Yang-Baxter equation in the infinite-dimensional Lie algebras from the Novikov algebras.

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