Non-Commutative Stochastic Independence: Algebraic and Analytic Aspects

该项目的目的是讨论独立性在非交换概率论中的作用，并研究与之密切相关的Lévy过程的概念。主要有两个方向：一方面，我们希望发展一个统一的处理大类的非交换独立的概念，现在正在研究或将在未来的研究。关于在不同公理集下的非交换独立性概念的分类问题也属于这一研究方向。另一方面，我们的目标是很好地理解最常见的非交换独立性概念，张量独立性，它对应于量子力学中的可观测量的独立性;具体地说，我们想解决的问题，分类生成的产品系统，研究辫子Hopf代数上的Lévy过程，并得到更明确的描述的Lévy过程的生成元紧量子群。

The aim of the project is to discuss the role of independence in non-commutative probability theory and to investigate the closely related concept of Lévy processes. There are two main directions: On the one hand, we wish to develop a unified treatment of large classes of notions of non-commutative independence which are studied now or will be studied in the future. Questions concerning the classification of notions of non-commutative independence under different sets of axioms also belong to this line of research. On the other hand, we aim at a good understanding of the most common notion of non-commutative independence, the tensor independence, which corresponds to independence of observables in quantum mechanics; specifically, we want to solve the problem of classifying the finitely generated product systems, study Lévy processes on braided Hopf algebras, and get more explicit descriptions of the generators of Lévy processes on compact quantum groups.