Toroidal actions on level 1 modules of U_q(\hat{sl_n})

Abstract

Recently Varagnolo and Vasserot established that the q-deformed Fock spaces due to Hayashi, and Kashiwara, Miwa and Stern, admit actions of the quantum toroidal algebra $U_q(sl_n,tor)$ (n > 2) with the level (0,1). In the present article we propose a more detailed proof of this fact then the one given by Varagnolo and Vasserot. The proof is based on certain non-trivial properties of Cherednik's commuting difference operators. The quantum toroidal action on the Fock space depends on a certain parameter. We find that with a specific choice of this parameter the action on the Fock spaces gives rise to the toroidal action on irreducible level-1 highest weight modules of the affine quantum algebra $U_q(\hat{sl}_n)$. Similarly, by a specific choice of the parameter, the level (1,0) vertex representation of the quantum toroidal algebra gives rise to a $U_q(sl_n,tor)$-module structure on irreducible level-1 highest weight $U_q(\hat{sl}_n)$-modules.