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

TL;DR

This paper constructs a new algebra from a subspace lattice with a flag, relates it to the quantum affine algebra $U_q(\,\widehat{\mathfrak{sl}}_2)$, and shows their modules are interconnected, extending known algebraic relations.

Contribution

It introduces an algebra $\,\mathcal{H}$ associated with a subspace lattice and a flag, establishing a homomorphism to the quantum affine algebra and linking their module structures.

Findings

Existence of an algebra homomorphism from $U_{q^{1/2}}(\,\widehat{\mathfrak{sl}}_2)$ to $\,\mathcal{H}$

Any irreducible $\,\mathcal{H}$-module remains irreducible as a $U_{q^{1/2}}(\,\widehat{\mathfrak{sl}}_2)$-module

Extension of the relation between incidence algebra and quantum algebra to the affine case

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.