A FAMILY OF NEW UNIVERSAL R-MATRICES

In this paper, we will study a class of quasitriangular Hopf algebras sl(q,z)(t)(2). Apart from the universal R-matrices, there are two othe r kinds of elements in sl(q,z)(t)(2) x sl(q,z)(2), which are called le ft universal R-matrices and right universal R-matrices, respectively. Both left universal R-matrices and right universal R-matrices are solu tions to the quantum Yang-Baxter equation. We find three varieties V-l , V-r and V, which have the properties: each point in V-l corresponds to a left universal R-matrix, each point in V-r corresponds to a right universal R-matrix and each point in V corresponds to a universal R-m atrix. Therefore, we obtain not only a family of new solutions to the quantum Yang-Baxter equation parametrized by the points in the variety V-l boolean OR V-r, but also a family of new universal R-matrices par ametrized by the points in the variety V. It is well known that the cl ass of 3-manifold invariants introduced by N. Yu. Reshetikhin and V. G . Turaev are a linear combination of certain colored framed link invar iants. The colored framed link invariants are special cases of the tan gle operators for colored framed tangles. Therefore, the tangle operat ors play an important role in the study of 3-manifold invariants. As a n application of the family of quasitriangular Hopf algebras, we will prove the existence of a class of tangle operators parametrized by the points in the variety V.