ABSTRACT

A quadratic Lie conformal algebra corresponds to a Hamiltonian pair in Gel'fend and Dorfman (Hamiltonian operators and algebraic structures related to them. Funkts Anal Prilozhen. 1979;13:13–30), which plays fundamental roles in completely integrable systems. Moreover, it also corresponds to a certain compatible pair of a Lie algebra and a Novikov algebra which was called Gel'fand–Dorfman bialgebra by Xu (Quadratic conformal superalgebras. J Algebra. 2000;231:1–38). 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.

摘要

二次李共形代数对应于 Gel'fend 和 Dorfman 中的哈密顿量对（哈密顿算子和与其相关的代数结构。Funkts Anal Prilozhen。1979；13：13-30），它在完全可积系统中起着基础作用。此外，它还对应于李代数和诺维科夫代数的某个相容对，徐称为 Gel'fand–Dorfman 双代数（Quadratic conformal superalgebras. J Algebra. 2000;231:1–38）。本文从Gel'fand-Dorfman双代数的角度研究了二次李共形代数的中心扩张和共形导数。结果表明，二次李共形代数的中心扩张和共形导子分别与相应的 Gel'fand–Dorfman 双代数的一些双线性形式和一些算子有关。