Representation theory of very non-standard quantum $so(2N-1)$

TL;DR

This paper classifies finite dimensional representations of a specific quantum symmetric pair algebra related to $so(2N-1)$, establishing a correspondence with dominant weights and showing semisimplicity under certain conditions.

Contribution

It provides a complete classification of simple modules for the quantum symmetric pair of type DII and links their characters to Weyl's formula, enabling Gelfand-Tsetlin bases construction.

Findings

One-to-one correspondence between simple modules and dominant weights.

Semisimplicity of the module category over fields of characteristic zero.

Characters given by Weyl's character formula.

Abstract

We classify the finite dimensional representations of the quantum symmetric pair coideal subalgebra $B_{\mathbf c}$ of type $DII$ corresponding to the symmetric pair $(so(2N),so(2N-1))$. For $B_{\mathbf c}$ defined over an arbitrary field $k$ and $q\in k$ not a root of unity we establish a one-to-one correspondence between finite dimensional, simple $B_{\mathbf c}$-modules and dominant integral weights for $so(2N-1)$. We use specialisation to show that the category of finite dimensional $B_{\mathbf c}$-modules is semisimple if $\mathrm{char}(k)=0$ and $q$ is transcendental over ${\mathbb Q}$. In this case the characters of simple $B_{\mathbf c}$-modules are given by Weyl's character formula. This means in particular that the quantum symmetric pair of type $DII$ can be used to obtain Gelfand-Tsetlin bases for irreducible representations of the Drinfeld-Jimbo quantum group $U_q(so(2N))$.