Mathematical Sciences: Analytical and Combinatorial Aspects of Subfactors

点击翻译按钮获取中文摘要

9531566 Bisch This is a project for mathematical research in the theory of operator algebras that deals, in particular, with analytical and combinatorial aspects of the Jones theory of subfactors. New hierarchies of finite-dimensional algebras, generalizing the Temperley-Lieb algebras in a natural way, will be used to study the structure of subfactors. These algebras capture, for instance, the symmetry coming from an intermediate subfactor and can be viewed as a certain minimal product of the standard invariant associated to two subfactors. A variety of new and interesting combinatorial structures appears naturally in this process. The technique of studying a subfactor through its intermediate subfactors is very natural, since it is on the classical level (i.e., the group level) just the analysis of a given group through its subgroups. It is planned to extend and refine the available intermediate subfactors techniques by studying the finite-dimensional algebras canonically associated to these subfactors. Popa systems, a certain natural system of inclusions of finite-dimensional algebras of symmetries, will play a key role in this analysis. Computational aspects of these symmetries will be examined, and it is planned to construct and analyze explicit examples of subfactors and commuting squares. Possible new interrelations of the theory of subfactors with mathematical physics and low-dimensional topology will be investigated. John von Neumann introduced his algebras of operators on a Hilbert space to study a variety of mathematical structures, some of which arise naturally in theoretical physics. In the early 80's, Vaughan Jones initiated a Galois theory for inclusions of certain von Neumann algebras and discovered that these inclusions are extremely rigid. They turn out to have a surprising braid group symmetry, which led Jones to the discovery of his famous knot invariant, the Jones polynomial. Besides these profound connections of the theory of subfactors to knot theory and low-dimensional topology, there are also deep relations to statistical mechanics, conformal field theory, and algebraic quantum field theory that have been beneficial to the understanding of naturally occurring structures in these areas. ***