Construction of Gel'fand-Dorfman Bialgebras from Classical R-Matrices

Super Hom-Gel'fand-Dorfman bialgebras and Hom-Lie conformal superalgebras

The purpose of this paper is to introduce and study super Hom-Gel'fand-Dorfman bialgebras and Hom-Lie conformal superalgebras. In this paper, we provide different ways for constructing super Hom-Gel'fand-Dorfman bialgebras and obtain some infinite-dimensional Hom-Lie superalgebras from affinization of super Hom-Gel'fand-Dorfman bialgebras. Also, we give a general construction of Hom-Lie conformal superalgebras from Hom-Lie superalgebras and establish equivalence of quadratic Hom-Lie conformal superalgebras and super Hom-Gel'fand-Dorfman bialgebras. Finally, we characterize one-dimensional cent…

Gel'fand-Dorfman Bialgebras

This is a survey talk on the study of Gel'fand-Dorfman bialgebras.

Gel'fand-Dorfman 代数的双代数理论及其在 Lie 共形双代数中的应用

A bialgebra theory of Gel'fand-Dorfman algebras with applications to Lie conformal bialgebras

本文定义 GD 双代数并用以构造 Lie 共形双代数，借助 GDYBE、$\mathcal{O}$-算子、pre-GD 代数、匹配对和 Manin 三元组建立其结构理论。

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…

Quadratic Lie conformal superalgebras related to Novikov superalgebras

We study quadratic Lie conformal superalgebras associated with No\-vikov superalgebras. For every Novikov superalgebra $(V,\circ)$, we construct an enveloping differential Poisson superalgebra $U(V)$ with a derivation $d$ such that $u\circ v = ud(v)$ and $\{u,v\} = u\circ v - (-1)^{|u||v|} v\circ u$ for $u,v\in V$. The latter means that the commutator Gelfand--Dorfman superalgebra of $V$ is special. Next, we prove that every quadratic Lie conformal superalgebra constructed on a finite-dimensional special Gel'fand--Dorfman superalgebra has a finite faithful conformal representation. This statem…

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.