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

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 $U_{q^{1\mathrm{/}2}}({\widehat{\mathfrak{s}\mathfrak{l}}}_2)$, 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 $U_{q^{1\mathrm{/}2}}({\mathfrak{s}\mathfrak{l}}_2)$. We show that there exists an algebra homomorphism from $U_{q^{1\mathrm{/}2}}({\widehat{\mathfrak{s}\mathfrak{l}}}_2)$ to $\mathcal{H}$ and that any irreducible module for $\mathcal{H}$ is irreducible as an $U_{q^{1\mathrm{/}2}}({\widehat{\mathfrak{s}\mathfrak{l}}}_2)$-module.