一类与Novikov代数相关的李共形代数的研究

李共形代数为研究顶点代数和满足局部条件的无限维李代数提供了强大的工具，且与非线性演化方程中的Hamiltonian 方法有紧密联系。绝大部分已知的李共形代数都是二次李共形代数，且该类代数与Gel'fand-Dorfman双代数一一对应。在本项目中，我们拟从Gel'fand-Dorfman双代数角度出发来统一地研究二次李共形代数的结构。主要刻画二次李共形代数的单性，共形导子等结构，研究这些结构与Gel'fand-Dorfman双代数某些结构的关系；建立Novikov代数与李共形代数的一般扩张理论，构建新的李共形双代数理论使其系数代数为一无限维李双代数，并研究二次李共形代数的扩张和双代数结构；刻画一类与Novikov代数相关的高维李共形代数的结构。以上研究不仅完善了李共形代数本身的理论，还为李共形代数和无限维李代数的研究提供了新的方法。此外，我们期待这些研究有广泛的物理前景。

Lie conformal algebras give us powerful tools for the study of vertex algebras and infinite-dimensional Lie algebras satisfying the locality property and have close connections to Hamiltonian formalism in the theory of nonlinear evolution equations. Most of Lie conformal algebras which are known are quadratic Lie conformal algebras which are one to one correspondence with Gel’fand-Dorfman bialgebras. In this project, we plan to study structures of quadratic Lie conformal algebras in a unified form in terms of Gel’fand-Dorfman bialgebras. We mainly characterize structures of quadratic Lie conformal algebras, such as simplicities, conformal derivations, explore the relationships between these structures and some structures of Gel’fand-Dorfman bialgebras, present the general extension theories of Novikov algebras and Lie conformal algebras, create a new theory of Lie conformal bialgebra whose coefficient algebra is an infinite-dimensional Lie bialgebra, study the extension problem and Lie conformal bialgebra structures of quadratic Lie conformal algebras, and characterize structures of a class of high dimensional Lie conformal algebras which are related with Novikov algebras. The above studies not only improve the theories of Lie conformal algebra, but also provide a new method to study Lie conformal algebras and infinite-dimensional Lie algebras. Moreover, we expect that these studies have a wide range of physical prospects.