An Algebra Associated with a Flag in a Subspace Lattice over a Finite Field and the Quantum Affine Algebra $U_q({\widehat{\mathfrak{sl}}}_2)$

Abstract

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.

arXiv comment40 pages

Cited by in corpus (1)

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