Extremal loop weight modules for $U_q({\widehat{sl}}_{\infty })$

Abstract

We construct by fusion product new irreducible representations of the quantum affinization $U_q({\widehat{sl}}_{\infty })$. The action is defined via the Drinfeld coproduct and is related to the crystal structure of semi-standard tableaux of type $A_{\infty }$. We call these representations extremal loop weight modules. The main motivations are applications to quantum toroidal algebras $U_q(s{l}_{n+1}^{tor})$: we prove the conjectural link between $U_q({\widehat{sl}}_{\infty })$ and $U_q(s{l}_{n+1}^{tor})$ stated in [14] for this family of representations. We recover in this way the extremal loop weight modules obtained in [23].

29 pages