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 U q (ŝl ∞ ). 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 U q (sl tor n+1 ): we prove the conjectural link between U q (ŝl ∞ ) and U q (sl tor n+1 ) stated in Hernandez (J. Algebra 329, [147][148][149][150][151][152][153][154][155][156][157][158][159][160][161][162] 2011) for these families of representations. We recover in this way the extremal loop weight modules obtained in Mansuy (arXiv:1305.3481).