Extremal loop weight modules and tensor products for quantum toroidal algebras

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

“…The main motivations are applications to quantum toroidal algebras Uq(sl tor n+1 ): we prove the conjectural link between Uq( ŝl∞) and Uq(sl tor n+1 ) stated in [14] for this family of representations. We recover in this way the extremal loop weight modules obtained in [23].…”

“…The construction of extremal loop weight modules is inspired by the original Kashiwara's study of the extremal weight modules (see [16]): we consider the fusion product of fundamental ℓ-highest weight modules and fundamental ℓ-lowest weight modules associated respectively to the fundamental weights Λ ℓ and −Λ 0 (ℓ ≥ 1) and to a non-zero complex parameter. The action of U q ( ŝl ∞ ) is defined by the Drinfeld coproduct, in the spirit of [23]. We show that we get in this way extremal loop weight modules associated to the weight Λ ℓ − Λ 0 , called extremal fundamental loop weight modules.…”

“…We prove the conjecture above for the family of extremal fundamental loop weight modules of U q ( ŝl ∞ ) of weight Λ ℓ −Λ 0 introduced in this paper: we determine q-character formulae for these representations. Furthermore, we show that their image by the morphism φ n is in fact the q-character of a representation of U q (sl tor n+1 ) constructed in [23]. Recall that our original goal is to construct extremal loop weight modules.…”

“…The quantum affinizations, in particular the quantum affine algebras and the quantum toroidal algebras, have been intensively studied (see for example [3,7,9,10,12,13,14,25] and references therein). In recent works [22,23] we constructed new families of integrable representations of the quantum toroidal algebra U q (sl tor n+1 ), called extremal loop weight modules, which generalize the ℓ-highest weight modules: there are representations generated by an extremal vector for the horizontal quantum affine subalgebra in the sense of Kashiwara [16,18]. The main motivation is the construction of finite-dimensional representations of the quantum toroidal algebra at roots of unity.…”

“…Second by using the combinatorial link with the quantum toroidal algebras, construct extremal loop weight modules for U q (sl tor n+1 ): we prove the conjecture above for the particular family of extremal fundamental loop weight modules. We recover in this way the extremal loop weight modules defined in [23] for the quantum toroidal algebras.…”