An algebra associated with a flag in a subspace lattice over a finite field and the quantum affine algebra

Yuta Watanabe¹

¹Tohoku university

The electronic journal of combinatorics, Tome 25 (2018) no. 4

Résumé

In this paper, we introduce an algebra $\mathcal{H}$ from a subspace lattice with respect to a fixed flag which contains its incidence algebra as a proper subalgebra. We then establish a relation between the algebra $\mathcal{H}$ and the quantum affine algebra $U_{q^{1/2}}(\widehat{\mathfrak{sl}}_2)$, where $q$ denotes the cardinality of the base field. It is an extension of the well-known relation between the incidence algebra of a subspace lattice and the quantum algebra $U_{q^{1/2}}(\mathfrak{sl}_2)$. We show that there exists an algebra homomorphism from $U_{q^{1/2}}(\widehat{\mathfrak{sl}}_2)$ to $\mathcal{H}$ and that any irreducible module for $\mathcal{H}$ is irreducible as an $U_{q^{1/2}}(\widehat{\mathfrak{sl}}_2)$-module.

Détail

DOI : 10.37236/7008

Classification : 51E20, 20G42
Mots-clés : subspace lattice, Young diagram, incidence algebra, quantum affine algebra