The $R$-matrix of the quantum toroidal algebra $U_{q,t}(\overset{..}{gl}_1)$ in the Fock module

Abstract

We propose a method to compute the $R$-matrix $R$ on a tensor product of Fock modules from coproduct relations in a Hopf algebra. We apply this method to the quantum toroidal algebra $U_{q,t}(\overset{..}{gl}_1)$ for which $R$ is currently not explicitly known. We show that the coproduct relations of $U_{q,t}(\overset{..}{gl}_1)$ reduce to a single elegant equation for $R$. Using the theory of symmetric Macdonald polynomials we show that this equation provides a recursive formula for the matrix elements of $R$.