Based modules over the $$\imath $$quantum group of type AI

Abstract

This paper studies classical weight modules over the $$\imath $$ quantum group $$\textbf{U}^{\imath }$$ of type AI. We introduce the notion of based $$\textbf{U}^{\imath }$$ -modules by generalizing the notion of based modules over quantum groups (quantized enveloping algebras). We prove that each finite-dimensional irreducible classical weight $$\textbf{U}^{\imath }$$ -module with integer highest weight is a based $$\textbf{U}^{\imath }$$ -module. As a byproduct, a new combinatorial formula for the branching rule from $$\mathfrak {sl}_n$$ to $$\mathfrak {so}_n$$ is obtained.