2 Yoshihisa Saito , Kouichi Takemura and Denis Uglov A

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(sln,tor) (n ≥ 3) 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 κ. 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 Uq(ŝln). 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(sln,tor)-module structure on irreducible level-1 highest weight Uq(ŝln)-modules.