对与约当代数相关的新代数结构的研究

Jordan algebras were introduced in 1930s by the physicist P. Jordan in an attempt to find an algebraic setting for quantum mechanics. In this project, we study the following new algebraic structures associated to Jordan algebras: pre-Jordan algebras, J-dendriform algebras, J-quadri-algebras and J-octo-algebras. And we mainly give a study on four topics associated to these algebras: (1)the relationship between two type of these algebras( That is, in which condition there exists a new algebraic structure in another algebraic structure)；（2）some equations (analogues of the Yang-Baxter equation on Lie algebras)on these algebras, and some special solutions to these equations；(3) symmetric or skew-symmetric bilinear forms on these algebras；(4)the relationship between one type of these algebras and other algebraic structure, such as Lie algebras. And these results are not only good for Jordan algebras, but also for their applications.

约当代数是一类很重要的非结合代数，最早于1930s由物理学家P.Jordan 提出来的。我们主要研究与约当代数相关的新的代数结构：预约当代数，J-dendriform 代数，J-quadri-代数，J-octo-代数。本项目主要主要研究上述代数的以下几个问题：（1）这几种代数结构之间的关系，指的是其中一个代数结构在什么条件下可以存在另外一种代数结构；(2)给出这些代数上的特定方程（类似于李代数上的Yang-Baxter 方程）,并给出它们的一些特殊解：（3）给出这些代数上的具有特定性质的对称或反对称的双线性型;(4)研究这些代数结构与其它代数结构，如李代数等的关系。这些工作的完成，一方面可以完善约当代数理论，另一方面也为约当代数在其它领域，如物理中的应用奠定了理论基础。