多八元数空间中若干问题研究

Octonion is a generalization of quaternion to nonassociative algebra which has closed relation with Yang-Mills equations, black hole, string theory and special relativity. Octonion analysis of several variables is a generalization of several complex variables to noncommutative and nonassociative field. Recently, analysis of octonion of several variables is in progressive development. An open problem arises to establish the solvability of the non-homogeneous Cauchy-Riemann equations with using complex geometry as well as Poincare-Bertrand formula in hypersurface. The purpose of this project is to solve the problem by constructing Cauchy-Riemann complex and applying analytic techniques from theory of several complex variables together with the technique dealing with non-commutativity in Clifford analysis. The new approach will enrich octonion analysis of several variables so that it will give new energy for the development of physics.

八元数是四元数的非结合推广与Yang-Mills 方程、黑洞、弦论和狭义相对论密切相关。本项目研究多八元数的分析理论，它是多复变函数论向非交换、非结合领域的推广。近年来，随着多八元数分析理论的不断完善，公开的问题是用复几何的方法研究非齐次Cauchy-Riemann方程、超曲面上Poincare-Bertrand公式。本项目针对上述问题，利用构造复形、Clifford分析中处理非交换的技巧以及八元数特有的结合子研究多八元数分析。它的成功实施将丰富八元数理论的研究内容和方法，为其在物理中的应用提供新鲜血液。

本项目致力于多八元数分析的研究，处于八元数分析、多复变函数论、数学物理的交叉领域。目前对八元数的研究几乎只限于向单复变的推广。本项目将多复变函数论向非交换、非结合的空间中推广，通过构造复形研究非齐次Cauchy-Riemann方程解的存在性定理，以及多八元数的积分理论如Bochner-Martinelli积分公式、不同曲面上的Plemelj公式和Poincare-Bertrand置换公式。这些研究成果丰富了多八元数分析的研究内容和方法,为多八元数分析本身以及物理中黑洞、弦理论、Yang-Mills方程提供良好的研究工具,从而奠定多八元分析自身的地位。