#### Vol. 267, No. 1, 2014

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Quantum extremal loop weight modules and monomial crystals

### Mathieu Mansuy

Vol. 267 (2014), No. 1, 185–241
##### Abstract

In this paper we construct a new family of representations for the quantum toroidal algebra ${\mathsc{U}}_{q}\left({sl}_{n+1}^{tor}\right)$, which are $\ell$-extremal in the sense of Hernandez. 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 this, we relate monomial realizations of level 0 extremal fundamental weight crystals to integrable representations of ${\mathsc{U}}_{q}\left({sl}_{n+1}^{tor}\right)$, 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.

##### Keywords
quantum toroidal algebras, extremal loop weight modules, monomial crystals, promotion operators
##### Mathematical Subject Classification 2010
Primary: 20G42, 81R10