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

Abstract

We classify the finite dimensional representations of the quantum symmetric pair coideal subalgebra $B_c$ of type DII corresponding to the symmetric pair $\left(\mathrm{so}\right(2N),\mathrm{so}(2N-1\left)\right)$. For $B_c$ defined over an arbitrary field k and $q\in k\setminus \left\{0\right\}$ not a root of unity we establish a one-to-one correspondence between finite dimensional, simple $B_c$-modules and dominant integral weights for $\mathrm{so}(2N-1)$. We use specialisation to show that the category of finite dimensional $B_c$-modules is semisimple if $\mathrm{char}\left(k\right)=0$ and q is transcendental over $Q$. In this case the characters of simple $B_c$-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 $U_q\left(\mathrm{so}\right(2N\left)\right)$.