The Askey–Wilson quotient conjecture for the centralizer of three $U_q(\mathfrak{sl}_2)$ representations

Let $j_1,j_2,j_3$ be representation labels, let $AW(3)$ be the centrally extended Askey–Wilson algebra, and let $\mathcal{C}_{j_1,j_2,j_3}$ be the centralizer under consideration. Define $\overline{AW}(j_1,j_2,j_3)$ as the quotient of $AW(3)$ by the relations fixing the central elements $\alpha_i=\chi_{j_i}$ and imposing the annihilating polynomial relations for $A$, $B$, $K$, $D$, and $D'$, together with

$$\Omega=\chi_{j_1}^2+\chi_{j_2}^2+\chi_{j_3}^2+K^2+\chi_{j_1}\chi_{j_2}\chi_{j_3}K-(q+q^{-1})^2.$$

Here $D$ and $D'$ are the elements defined from the Askey–Wilson generators, and $\Omega$ is the central element of $AW(3)$. The Askey–Wilson quotient conjecture. The map

$$\overline{\phi}:\overline{AW}(j_1,j_2,j_3)\longrightarrow\mathcal{C}_{j_1,j_2,j_3},\qquad \overline{\phi}(A)=C_{12},\quad \overline{\phi}(B)=C_{23},\quad \overline{\phi}(K)=C_{123},$$

is an algebra isomorphism. This conjecture proposes a presentation of the centralizer as a finite quotient of the centrally extended Askey–Wilson algebra; the supplied text gives no resolution status.