Toroidal actions on level 1 modules ofU_q (widehat{mathfrak{s}mathfrak{l}}_n )

Recently Varagnolo and Vasserot established that theq-deformed Fock spaces due to Hayashi, and Kashiwara, Miwa and Stern, admit actions of the quantum toroidal algebra with the level (0,1). In the present article we propose a more detailed proof of this fact than the one given by Varagnolo and Vasserot. 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. Similarly, by a specific choice of the parameter, the level (1,0) vertex representation of the quantum totoidal algebra gives rise to a structure on irreducible level-1 highest weight-modules.Supported by the JSPS Research Fellowship for Young Scientists.