8 problems

Let ${\bf K}^{(\frac{1}{2})}(u)$ and ${\cal K}^{(\frac{1}{2})}(u)$ be, respectively, the spin-$\frac{1}{2}$ specialization of a universal K-matrix and the fused K-operator, with…

Let $O_q$ be the $q$-Onsager algebra, and let $\tilde G_{k+1}$, $k\in\mathbb N$, be its alternating generators. For $x\in O_q$, consider the subalgebra generated by…

Let $O_q$ be the $q$-Onsager algebra, with alternating generators $\{W_{-i}\}_{i\in\mathbb N}$, $\{\tilde G_{j+1}\}_{j\in\mathbb N}$, and $\{W_{k+1}\}_{k\in\mathbb N}$. A linear or…

Let $\mathcal A_q$ be the current algebra, let $\mathcal O_q$ be the q-Onsager algebra with generators $A,B$, let $\{\tilde g_{k+1}\}_{k\in\mathbb N}\subseteq\mathcal J$ be the seq…

Let $\mathcal O_q$ be the q-Onsager algebra with generators $A,B$, and let $\mathcal J$ be its two-sided ideal generated by $A$ and $B$. Lower-generator uniqueness conjecture. Ther…

Let $\mathcal A_q$ be the current algebra, let $Z(\mathcal A_q)$ denote its center, let $\Delta_{k+1}\in Z(\mathcal A_q)$, and let $\langle\mathcal W_0,\mathcal W_1\rangle$ be the…

Bijection conjecture. The map $\Psi$ is a bijection.

PBW basis conjecture. These vectors induce a Poincare–Birkhoff–Witt-type basis of $\tilde{{\cal A}}_q^{\{\delta\}}$, called the $WG$-basis.