arXiv subjects

Yanyong Hong

Publications and source records attributed to Yanyong Hong.

45 records · Page 3Linked to original sources

arXiv2018

Conformal classical Yang-Baxter equation, $S$-equation and $\mathcal{O}$-operators

Conformal classical Yang-Baxter equation and $S$-equation naturally appear in the study of Lie conformal bialgebras and left-symmetric conformal bialgebras. In this paper, they are interpreted in terms of a kind of operators, namely, $\mathcal O$-operators in the conformal sense. Explicitly, the skew-symmetric part of a conformal linear map $T$ where $T_0=T_λ\mid_{λ=0}$ is an $\mathcal O$-operator in the conformal sense is a skew-symmetric solution of conformal classical Yang-Baxter equation, whereas the symmetric part is a symmetric solution of conformal $S$-equation. One byproduct is that a finite left-symmetric conformal algebra which is a free $\mathbb{C}[\partial]$-module gives a natural $\mathcal O$-operator and hence there is a construction of solutions of conformal classical Yang-Baxter equation and conformal $S$-equation from the former. Another byproduct is that the non-degenerate solutions of these two equations correspond to 2-cocycles of Lie conformal algebras and left-symmetric conformal algebras respectively. We also give a further study on a special class of $\mathcal{O}$-operators called Rota-Baxter operators on Lie conformal algebras and some explicit examples are presented.

math.RA↗

arXiv2017

Extending structures for associative conformal algebras

In this paper, we give a study of the $\mathbb{C}[\partial]$-split extending structures problem for associative conformal algebras. Using the unified product as a tool, which includes interesting products such as bicrossed product, cocycle semi-direct product and so on, a cohomological type object is constructed to characterize the $\mathbb{C}[\partial]$-split extending structures for associative conformal algebras. Moreover, using this theory, the extending structures of an associative conformal algebra $A$ which is free as a $\mathbb{C}[\partial]$-module by the $\mathbb{C}[\partial]$-module $Q=\mathbb{C}[\partial]x$ are described using flag datums of $A$. Furthermore, we give a classification of the extending structures of $A$ by $Q=\mathbb{C}[\partial]x$ in detail up to equivalence when $A$ is a free associative conformal algebra of rank 1.

math.RA↗

arXiv2016

A family of new simple modules over the Schrödinger-Virasoro algebra

In this article, a large class of simple modules over the Schrödinger-Virasoro algebra $\mathcal{G}$ are constructed, which include highest weight modules and Whittaker modules. These modules are determined by the simple modules over the finite-dimensional quotient algebras of some subalgebras. Moreover, we show that all simple modules of $\mathcal{G}$ with locally finite actions of elements in a certain positive part belong to this class of simple modules. Similarly, a large class of simple modules over the $W$-algebra $W(2,2)$ are constructed.

math.RT↗

arXiv2016

Generalized conformal derivations of Lie conformal algebras

Let $R$ be a Lie conformal algebra. The purpose of this paper is to investigate the conformal derivation algebra $CDer(R)$, the conformal quasiderivation algebra $QDer(R)$ and the generalized conformal derivation algebra $GDer(R)$. The generalized conformal derivation algebra is a natural generalization of the conformal derivation algebra. Obviously, we have the following tower $CDer(R)\subseteq QDer(R)\subseteq GDer(R)\subseteq gc(R)$, where $gc(R)$ is the general Lie conformal algebra. Furthermore, we mainly research the connection of these generalized conformal derivations. Finally, the conformal $(α,β,γ)$-derivations of Lie conformal algebras are studied. Moreover, we obtain some connections between several specific generalized conformal derivations and the conformal $(α,β,γ)$-derivations. In addition, all conformal $(α,β,γ)$-derivations of finite simple Lie conformal algebras are characterized.

math.QA↗

arXiv2015

Extending structures and classifying complements for left-symmetric algebras

Let $A$ be a left-symmetric (resp. Novikov) algebra, $E$ be a vector space containing $A$ as a subspace and $V$ be a complement of $A$ in $E$.The extending structures problem which asks for the classification of all left-symmetric (resp. Novikov) algebra structures on $E$ such that $A$ is a subalgebra of $E$ is studied. In this paper, the definition of the unified product of left-symmetric (resp. Novikov) algebras is introduced. It is shown that there exists a left-symmetric (resp. Novikov) algebra structure on $E$ such that $A$ is a subalgebra of $E$ if and only if $E$ is isomorphic to a unified product of $A$ and $V$. Two cohomological type objects $\mathcal{H}_A^2(V,A)$ and $\mathcal{H}^2(V,A)$ are constructed to give a theoretical answer to the extending structures problem. Furthermore, given an extension $A\subset E$ of left-symmetric (resp. Novikov) algebras, another cohomological type object is constructed to classify all complements of $A$ in $E$. Several special examples are provided in details.

math.RA↗

arXiv2015

On Schrödinger-Virasoro type Lie conformal algebras

In this paper, two new classes of Schrödinger-Virasoro type Lie conformal algebras TSV(a,b) and TSV(c) which are non-simple are introduced for some a, b, c\in \mathbb{C}. Moreover, central extensions, conformal derivations and free conformal modules of rank 1 of TSV(a,b) and TSV(c) are determined.

math.QA↗

arXiv2015

Simplicity of quadratic Lie conformal algebras

In this paper, simplicity of quadratic Lie conformal algebras are investigated. From the point view of the corresponding Gel'fand-Dorfman bialgebras, some sufficient conditions and necessary conditions to ensure simplicity of quadratic Lie conformal algebras are presented. By these observations, we present several classes of new infinite simple Lie conformal algebras. These results will be useful for classification purposes.

math.QA↗

arXiv2014

A class of Lie conformal superalgebras in higher dimensions

Fix a positive integer number $r$. A class of $r$-dim Lie conformal superalgebras named $r$-dim $i$-linear Lie conformal superalgebras are studied for $1\leq i \leq r$. We present an equivalent characterization of this class of Lie conformal superalgebras in $r$ dimension. Moreover, by this equivalent characterization, several constructions and examples are given.

math.QA↗

arXiv2013

Analysis of Uq(sl(m+1))-symmetries on quantum n-spaces

In this paper, the module-algebra structures of $U_q(sl(m+1))$ on the quantum $n$-space $A_q(n)$ are studied. We characterize all module-algebra structures of $U_q(sl(m+1))$ on $A_q(2)$ and $A_q(3)$ when $m\geq 2$. The module-algebra structures of $U_q(sl(m+1))$ on $A_q(n)$ are also considered for any $n\geq 4$.

math.QA↗