AbasisfortheBirman-Wenzl创新.PDFVIP

A basis for the Birman-Wenzl algebra H. R. Morton and A. J. Wassermann Foreword (2000) This paper is a very lightly edited version of an article originally written in 1989 but never fully completed. We had planned a final section to make use of the ability to change at will between the Birman-Wenzl algebra, as given by generators and relations, and the geometric framework of the tangles, so as to look in more detail at the representation theory. One goal of our original approach was to make sure that specialisations of the coefficient ring could be handled confidently, and that the translations to and from the tangle context were on a sound footing. More recently others have made progress in this way, in works such as [6], but we have had a number of requests for our earlier account, and so we have put it into this more accessible form. 1 Introduction (1989) In recent years there has been considerable interest in deformations of the classical ‘centraliser algebras’ of Schur, Weyl and Brauer. These play an important role in several areas, including exactly solvable models in statistical mechanics, quantum groups, von Neumann algebras and knot theory. It has long been recognized that these links are more than tenuous and if properly exploited lead to fruitful interchanges between the different disciplines. The first and most spectacular instance of this was of course Vaughan Jones’ pioneering work on subfactors, which led to his discovery of new link invariants. Subsequently these invariants were understood in terms of solutions of the quantum Yang-Baxter equation and vertex models. The central thread running through all these topics is the quantum group obtained by deforming the universal enveloping algebra of the unitary group. One has also to defo