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Yanyong Hong

Publications and source records attributed to Yanyong Hong.

At least 37 records · Page 2Linked to original sources
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.

math.QA↗
arXiv2022

Finite irreducible modules of a class of $\mathbb{Z}^+$-graded Lie conformal algebras

In this paper, we introduce the notion of completely non-trivial module of a Lie conformal algebra. By this notion, we classify all finite irreducible modules of a class of $\mathbb{Z}^+$-graded Lie conformal algebras $\mathcal{L}=\bigoplus_{i=0}^{\infty} \mathbb{C}[\partial]L_i$ satisfying $ [{L_0}_λL_0]=(\partial+2λ)L_0,$ and $[{L_1}_λL_i]\neq 0$ for any $i\in \mathbb{Z}^+$. These Lie conformal algebras include Block type Lie conformal algebra $\mathcal{B}(p)$ and map Virasoro Lie conformal algebra $\mathcal{V}(\mathbb{C}[T])=Vir\otimes \mathbb{C}[T]$. As a result, we show that all non-trivial finite irreducible modules of these algebras are free of rank one as a $\mathbb{C}[\partial]$-module.

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arXiv2022

New irreducible non-weight Virasoro modules from tensor products

In this paper, we present a class of non-weight Virasoro modules $\mathcal{M}\big(V,Ω(λ_0,α_0)\big)\otimes\bigotimes_{i=1}^mΩ(λ_i,α_i)$ where $Ω(λ_i,α_i)$ and $\mathcal{M}\big(V,Ω(λ_0,α_0)\big)$ are irreducible Virasoro modules defined in \cite{LZ2} and \cite{LZ} respectively. The necessary and sufficient conditions for $\mathcal{M}\big(V,Ω(λ_0,α_0)\big)\otimes\bigotimes_{i=1}^mΩ(λ_i,α_i)$ to be irreducible are obtained. Then we determine the necessary and sufficient conditions for two such irreducible Virasoro modules to be isomorphic. At last, we show that the irreducible modules in this class are new.

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arXiv2022

Extending structures for Gel'fand-Dorfman bialgebras

Gel'fand-Dorfman bialgebra, which is both a Lie algebra and a Novikov algebra with some compatibility condition, appears in the study of Hamiltonian pairs in completely integrable systems and a class of special Lie conformal algebras called quadratic Lie conformal algebras. In this paper, we investigate the extending structures problem for Gel'fand-Dorfman bialgebras, which is equivalent to some extending structures problem of quadratic Lie conformal algebras. Explicitly, given a Gel'fand-Dorfman bialgebra $(A, \circ, [\cdot,\cdot])$, this problem asks that how to describe and classify all Gel'fand-Dorfman bialgebraic structures on a vector space $E$ $(A\subset E$) such that $(A, \circ, [\cdot,\cdot])$ is a subalgebra of $E$ up to an isomorphism whose restriction on $A$ is the identity map. Motivated by the theories of extending structures for Lie algebras and Novikov algebras, we construct an object $\mathcal{GH}^2(V,A)$ to answer the extending structures problem by introducing a definition of unified product for Gel'fand-Dorfman bialgebras, where $V$ is a complement of $A$ in $E$. In particular, we investigate the special case when $\text{dim}(V)=1$ in detail.

math.RA↗
arXiv2021

Extending structures for Lie bialgebras

Let $(\mathfrak{g}, [\cdot,\cdot], δ_\mathfrak{g})$ be a fixed Lie bialgebra, $E$ be a vector space containing $\mathfrak{g}$ as a subspace and $V$ be a complement of $\mathfrak{g}$ in $E$. A natural problem is that how to classify all Lie bialgebraic structures on $E$ such that $(\mathfrak{g}, [\cdot,\cdot], δ_\mathfrak{g})$ is a Lie sub-bialgebra up to an isomorphism of Lie bialgebras whose restriction on $\mathfrak{g}$ is the identity map. This problem is called the extending structures problem. In this paper, we introduce a general co-product on $E$, called the unified co-product of $(\mathfrak{g},δ_\mathfrak{g})$ by $V$. With this unified co-product and the unified product of $(\mathfrak{g}, [\cdot,\cdot])$ by $V$ developed in \cite{AM1}, the unified bi-product of $(\mathfrak{g}, [\cdot,\cdot], δ_\mathfrak{g})$ by $V$ is introduced. Moreover, we show that any $E$ in the extending structures problem is isomorphic to a unified bi-product of $(\mathfrak{g}, [\cdot,\cdot], δ_\mathfrak{g})$ by $V$. Then an object $\mathcal{HBI}_{\mathfrak{g}}^2(V,\mathfrak{g})$ is constructed to classify all $E$ in the extending structures problem. Moreover, several special unified bi-products are also introduced. In particular, the unified bi-products when $\text{dim} V=1$ are investigated in detail.

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arXiv2021

On antisymmetric infinitesimal conformal bialgebras

In this paper, we construct a bialgebra theory for associative conformal algebras, namely antisymmetric infinitesimal conformal bialgebras. On the one hand, it is an attempt to give conformal structures for antisymmetric infinitesimal bialgebras. On the other hand, under certain conditions, such structures are equivalent to double constructions of Frobenius conformal algebras, which are associative conformal algebras that are decomposed into the direct sum of another associative conformal algebra and its conformal dual as $\mathbb{C}[\partial]$-modules such that both of them are subalgebras and the natural conformal bilinear form is invariant. The coboundary case leads to the introduction of associative conformal Yang-Baxter equation whose antisymmetric solutions give antisymmetric infinitesimal conformal bialgebras. Moreover, the construction of antisymmetric solutions of associative conformal Yang-Baxter equation is given from $\mathcal{O}$-operators of associative conformal algebras as well as dendriform conformal algebras.

math.QA↗
arXiv2021

Finite irreducible conformal modules of rank two Lie conformal algebras

In the present paper, we prove that any finite non-trivial irreducible module over a rank two Lie conformal algebra $\mathcal{H}$ is of rank one. We also describe the actions of $\mathcal{H}$ on its finite irreducible modules explicitly. Moreover, we show that all finite non-trivial irreducible modules of finite Lie conformal algebras whose semisimple quotient is the Virasoro Lie conformal algebra are of rank one.

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arXiv2021

Finite irreducible conformal modules over the Lie conformal superalgebra $\mathcal{S}(p)$

In the present paper, we introduce a class of infinite Lie conformal superalgebras $\mathcal{S}(p)$, which are closely related to Lie conformal algebras of extended Block type defined in \cite{CHS}. Then all finite non-trivial irreducible conformal modules over $\mathcal{S}(p)$ for $p\in\C^*$ are completely classified. As an application, we also present the classifications of finite non-trivial irreducible conformal modules over finite quotient algebras $\mathfrak{s}(n)$ for $n\geq1$ and $\mathfrak{sh}$ which is isomorphic to a subalgebra of Lie conformal algebra of $N=2$ superconformal algebra. Moreover, as a generalized version of $\mathcal{S}(p)$, the infinite Lie conformal superalgebras $\mathcal{GS}(p)$ are constructed, which have a subalgebra isomorphic to the finite Lie conformal algebra of $N=2$ superconformal algebra.

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arXiv2020

Classifying complements for conformal algebras

Let $R\subseteq E$ be two Lie conformal algebras and $Q$ be a given complement of $R$ in $E$. Classifying complements problem asks for describing and classifying all complements of $R$ in $E$ up to an isomorphism. It is known that $E$ is isomorphic to a bicrossed product of $R$ and $Q$. We show that any complement of $R$ in $E$ is isomorphic to a deformation of $Q$ associated to the bicrossed product. A classifying object is constructed to parameterize all $R$-complements of $E$. Several explicit examples are provided. Similarly, we also develop a classifying complements theory of associative conformal algebras.

math.RA↗
arXiv2020

Finite irreducible conformal modules over the extended Block type Lie conformal algebra $\mathfrak{B}(α,β,p)$

In this paper, we introduce a class of infinite Lie conformal algebras $\mathfrak{B}(α,β,p)$, which are the semi-direct sums of Block type Lie conformal algebra $\mathfrak{B}(p)$ and its non-trivial conformal modules of $\Z$-graded free intermediate series. The annihilation algebras are a class of infinite-dimensional Lie algebras, which include a lot of interesting subalgebras: Virasoro algebra, Block type Lie algebra, twisted Heisenberg-Virasoro algebra and so on. We give a complete classification of all finite non-trivial irreducible conformal modules of $\mathfrak{B}(α,β,p)$ for $α,β\in\C, p\in\C^*$. As an application, the classifications of finite irreducible conformal modules over a series of finite Lie conformal algebras $\mathfrak{b}(n)$ for $n\geq1$ are given.

math.RT↗
arXiv2019

Central extensions and conformal derivations of a class of Lie conformal algebras

A quadratic Lie conformal algebra corresponds to a Hamiltonian pair in \cite{GD}, which plays fundamental roles in completely integrable systems. Moreover, it also corresponds to certain compatible pairs of a Lie algebra and a Novikov algebra which was called Gel'fand-Dorfman bialgebra by Xu in \cite{X1}. In this paper, central extensions and conformal derivations of quadratic Lie conformal algebras are studied in terms of Gel'fand-Dorfman bialgebras. It is shown that central extensions and conformal derivations of a quadratic Lie conformal algebra are related with some bilinear forms and some operators of the corresponding Gel'fand-Dorfman bialgebra respectively.

math.QA↗
arXiv2019

Extensions of finite irreducible modules of Lie conformal algebras $\mathcal{W}(a,b)$ and some Schrödinger-Virasoro type Lie conformal algebras

Lie conformal algebras $\mathcal{W}(a,b)$ are the semi-direct sums of Virasoro Lie conformal algebra and its nontrivial conformal modules of rank one. In this paper, we give a complete classification of extensions of finite irreducible conformal modules of $\mathcal{W}(a,b)$. With a similar method, we characterize all extensions of finite irreducible conformal modules of Schrödinger-Virasoro type Lie conformal algebras $TSV(a,b)$ and $TSV(c)$.

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arXiv2019

On a class of infinite simple Lie conformal algebras

In this paper, we study a class of infinite simple Lie conformal algebras associated to a class of generalized Block type Lie algebras. The central extensions, conformal derivations and free intermediate series modules of this class of Lie conformal algebras are determined. Moreover, we also show that these Lie conformal algebras do not have any non-trivial finite conformal modules. Consequently, these Lie conformal algebras cannot be embedded into $gc_N$ for any positive integer $N$.

math.RT↗
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↗
arXiv2019

Finite irreducible modules of Lie conformal algebras $\mathcal{W}(a,b)$ and some Schrödinger-Virasoro type Lie conformal algebras

Lie conformal algebras $\mathcal{W}(a,b)$ are the semi-direct sums of Virasoro Lie conformal algebra and its nontrivial conformal modules of rank one. In this paper, we first give a complete classification of all finite nontrivial irreducible conformal modules of $\mathcal{W}(a,b)$. It is shown that all such modules are of rank one. Moreover, with a similar method, all finite nontrivial irreducible conformal modules of Schrödinger-Virasoro type Lie conformal algebras $TSV(a,b)$ and $TSV(c)$ are characterized.

math.QA↗
arXiv2018

Quadratic Leibniz conformal algebras

In this paper, we study a class of Leibniz conformal algebras called quadratic Leibniz conformal algebras. An equivalent characterization of a Leibniz conformal algebra $R=\mathbb{C}[\partial]V$ through three algebraic operations on $V$ are given. By this characterization, several constructions of quadratic Leibniz conformal algebras are presented. Moreover, one-dimensional central extensions of quadratic Leibniz conformal algebras are considered using some bilinear forms on $V$. In particular, we also study one-dimensional Leibniz central extensions of quadratic Lie conformal algebras.

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