A Variety of Solutions to the Yang-Baxter Equation

Abstract

In this note the author gives a complete description of allinvertible solutions of the Yang Baxter equations on V\otimes Vthat are of the form σ+P, where σ is the flip andP is a rank one transformation. He shows that the set of suchP with Pe0 and R=σ+P invertible forms an algebraicvariety of dimension n\sp 2+[n/2]. He also shows how toconstruct almost all solutions to the Yang Baxter equation ofthis type. Unfortunately, the solutions of the Yang Baxterequations of the form studied in this note ``only'' produce theJones polynomial. At last, he shows how an open subset of thesesolutions gives rise to representations of Temperley Liebalgebras.\par {For the entire collection see MR 99h:00015.}