Quantum extremal loop weight modules and monomial crystals

Abstract

In this paper we construct a new family of representations for the quantum toroidal algebras of type $A_n$, which are $\ell$-extremal in the sense of Hernandez [24]. We construct extremal loop weight modules associated to level 0 fundamental weights $\varpi_\ell$ when $n=2r+1$ is odd and $\ell=1, r+1$ or $n$. To do it, we relate monomial realizations of level 0 extremal fundamental weight crystals with integrable representations of $\mathcal{U}_q(sl_{n+1}^{tor})$, and we introduce promotion operators for the level 0 extremal fundamental weight crystals. By specializing the quantum parameter, we get finite-dimensional modules of quantum toroidal algebras at roots of unity. In general, we give a conjectural process to construct extremal loop weight modules from monomial realizations of crystals.Comment: 49 pages. Accepted for publication in Pacific Journal of Mathematic

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

“…In this section, we prove [12,Conjecture 5.3] for the family of extremal fundamental loop weight modules of U q (ŝl ∞ ): we recover in this way the extremal fundamental loop weight U q (sl tor n+1 )-modules constructed in [19,20]. In the first part, we recall some definitions about the quantum toroidal algebras U q (sl tor n+1 ) (n ≥ 2) and its conjectural link with U q (ŝl ∞ ).…”

“…Proof This result is known for the fundamental U q (ŝl n+1 )-modules (see [20,Theorem 4.12]). The proposition follows by inductive limit, using Theorem 2.15.…”

“…The definition of extremal loop weight modules is given in [12] for the quantum toroidal algebras and studied in [19,20]. We introduce its definition in the context of the quantum affinization U q (ŝl ∞ ).…”

“…They are quantum groups analogs of elliptic Cherednik algebras [3] to whom they are related via Schur-Weyl duality [23]. The author has defined extremal loop weight modules in the context of U q (sl tor n+1 ) [19,20]. The main application is the construction of finite-dimensional representations of the quantum toroidal algebras at roots of unity.…”

“…In recent works [19,20] 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 -highest weight modules: these are representations generated by an extremal vector for the horizontal quantum affine subalgebra in the sense of Kashiwara [13,15]. The main application is the construction of finite-dimensional representations of quantum toroidal algebras at roots of unity.…”