This wide-ranging project is mathematical research that originates in the theory of operator algebras, with collateral impact in several other areas of mathematics. Algebras of operators on Hilbert space have long been used to represent a diversity of mathematical structures, including some that arise in theoretical physics. Professor Wenzl will pursue several consequences of a deceptively simple construction for building towers of increasingly larger operator algebras of a certain special sort. More specifically, he will continue his research into Hecke algebras and a recently discovered class of algebras connected with the Kauffman link invariant in knot theory. He will explore the connections these algebras have to representations of Lie groups. More examples of subfactors of the hyperfinite factor studied by operator algebraists are expected to be found. Related issues in mathematical physics, for instance the quantum Yang - Baxter equations, will be investigated.