Toroidal actions on level 1 modules of $$U_q (\widehat{\mathfrak{s}\mathfrak{l}}_n )$$

Abstract

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\(U'_q (\mathfrak{s}\mathfrak{l}_{n,tor} ) (n \geqslant 3)\) 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 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\(U_q (\widehat{\mathfrak{s}\mathfrak{l}}_n )\). Similarly, by a specific choice of the parameter, the level (1,0) vertex representation of the quantum totoidal algebra gives rise to a\(U'_q (\mathfrak{s}\mathfrak{l}_{n,tor} ) (n \geqslant 3)\) structure on irreducible level-1 highest weight\(U_q (\widehat{\mathfrak{s}\mathfrak{l}}_n )\)-modules.