Computing the R-matrix of the quantum toroidal algebra

Abstract

We consider the problem of the R-matrix of the quantum toroidal algebra Uq,t(gl..1) in the Fock representation. Using the connection between the R-matrix R(u) (u being the spectral parameter) and the theory of Macdonald operators, we obtain explicit formulas for R(u) in the operator and matrix forms. These formulas are expressed in terms of the eigenvalues of a certain Macdonald operator, which completely describe the functional dependence of R(u) on the spectral parameter u. We then consider the geometric R-matrix (obtained from the theory of K-theoretic stable bases on moduli spaces of framed sheaves), which is expected to coincide with R(u) and thus gives another approach to the study of the poles of the R-matrix as a function of u.