Extremal loop weight modules for Uq(sl∞)

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We construct by fusion product new irreducible representations of the quantum affinization Uq(sl∞). 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 Uq(slₙ₊₁tor): we prove the conjectural link between Uq(sl∞) and Uq(slₙ₊₁tor) stated in [14] for this family of representations. We recover in this way the extremal loop weight modules obtained in [23].