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 ${\mathcal{U}}_q(s{l}_{n+1}^{tor})$, 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.

49 pages. Accepted for publication in Pacific Journal of Mathematics

References in corpus (4)

• Quantum toroidal ${\mathfrak{gl}}_1$ algebra : plane partitions
• Existence of Kirillov-Reshetikhin crystals for nonexceptional types
• Combinatorial structure of Kirillov-Reshetikhin crystals of type D_n(1), B_n(1), A_{2n-1}(2)
• Quantum loop modules