Extremal Loop Weight Modules for U q ( sl ̂ ∞ ) $\mathcal {U}_{q}(\hat {sl}_{\infty })$

Abstract

We construct by fusion product new families of irreducible representations of the quantum affinization $\mathcal {U}_{q}(\hat {sl}_{\infty })$ . The action is defined via the Drinfeld coproduct and is related to the crystal structure of semi-standard tableaux of type A ∞ . We call these representations extremal loop weight modules. The main motivations are applications to quantum toroidal algebras $\mathcal {U}_{q}(sl_{n+1}^{tor})$ : we prove the conjectural link between $\mathcal {U}_{q}(\hat {sl}_{\infty })$ and $\mathcal {U}_{q}(sl_{n+1}^{tor})$ stated in Hernandez (J. Algebra 329, 147–162, 2011) for these families of representations. We recover in this way the extremal loop weight modules obtained in Mansuy (arXiv: 1305.3481 ).