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lsamvy过程密切相关的概念。主要有两个方向：一方面，我们希望对现在正在研究或将来将要研究的大量非交换独立性概念发展一种统一的处理方法。关于不同公理集下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非交换独立性的最常见概念，张量独立性，它对应于量子力学中可观测物的独立性；具体地说，我们想要解决有限生成积系统的分类问题，研究编织Hopf代数上的lsamvy过程，并得到紧量子群上lsamvy过程的生成器的更明确的描述。

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.