Based modules over the $\mathrm{ı}$quantum group of type AI

This paper studies classical weight modules over the $\mathrm{ı}$quantum group ${\mathbf{U}}^{\mathrm{ı}}$ of type AI. We introduce the notion of based ${\mathbf{U}}^{\mathrm{ı}}$-modules by generalizing the notion of based modules over the quantum groups. We prove that each finite-dimensional irreducible classical weight ${\mathbf{U}}^{\mathrm{ı}}$-module with integer highest weight is a based ${\mathbf{U}}^{\mathrm{ı}}$-module. As a byproduct, a new combinatorial formula for the branching rule from ${\mathfrak{s}\mathfrak{l}}_n$ to ${\mathfrak{s}\mathfrak{o}}_n$ is obtained.