25 problems

Let $\mathcal A_q$ be the current algebra and let $\mathcal O_q$ be the $q$-Onsager algebra. Baseilhac–Belliard's homomorphic-image conjecture. The algebra $\mathcal O_q$ is a homo…

Let $A$ and $A^*$ be the fundamental generators of the $q$-Onsager algebra, with defining parameters $\rho_0$ and $\rho_1$. For a nonnegative integer $r$, define the coefficients…

WG-basis conjecture. The ordered monomials in the three blocks $W^-<G<W^+$ form a PBW basis; this is the $WG$-basis.

Let $O_q$ be the $q$-Onsager algebra and let $mathcal A_q$ be the alternating central extension associated with it. Alternating central-extension conjecture. On the basis of suppor…

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 $\cA_q$ be the alternating central extension of the q-Onsager algebra, and let $V$ be an evaluation representation of $U_q(\widehat{\mathfrak{sl}}_2)$. Consider the certain exp…

Uniqueness conjecture. There is no homomorphism apart from the one already given that sends $\mathcal W_0$ and $\mathcal W_1$ to elements of this form.

Spin-$j$ solution conjecture. Spin-$j$ fused K-operator solutions are unique up to an overall scalar factor, their matrix entries are linear combinations of monomials of the prescr…

Universal K-matrix evaluation conjecture. For every $j\in\frac{1}{2}\mathbb{N}$,

Arbitrary-spin uniqueness conjecture. The twisted intertwining relations determine uniquely the K-operators for any $j$, and these K-operators match the fused K-operators construct…

Universal K-matrix existence conjecture. A universal K-matrix $\mathfrak{K}$ for ${\cal A}_q$ satisfying the paper's axioms with the choice $\psi=\eta$ exists.

Uniqueness and fusion conjecture. In general, solutions of the twisted intertwining relations are uniquely determined and match the expressions of the fused K-operators.

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…

Baseilhac–Belliard's generator conjecture. There exist elements in $\mathcal O_q$ belonging to these four families that satisfy the defining relations for $\mathcal A_q$.

Baseilhac–Belliard's conjecture. The homomorphism

Let ${\cal A}_q^S$ denote the Sklyanin reflection-algebra presentation of the algebra ${\cal A}_q$, and let $O_q$ be the $q$-Onsager algebra. Central-extension conjecture. The alge…

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…

Let $\mathcal O_q$ be the $q$-Onsager algebra with generators $A$, $B$, and let $C=(q^{-1}BA-qAB)/(q^2-q^{-2})$. Cyclic automorphism conjecture. There exists an automorphism of…

Let $\cb_c$ be the $q$-Onsager algebra over $\Q(q)$, depending on a parameter $c\in\Q(q)$, and let ${\mathcal A}_q$ be the current algebra proposed in the cited work. Current algeb…

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

Let $N\geq 2$, let $q=e^{\frac{\mathrm{i}\pi}{N}}$, and consider the algebra generated by the four operators $\W_i$ and $\NtcWi$, $i\in\mathbb{Z}_2$, subject to relations (1)–(4):…

Let $\widehat{g}$ be a simply-laced affine Lie algebra, and let $\mathcal O_q(\widehat{g})$ be the generalized $q$-Onsager algebra with generators $A_i$ and parameters $\rho_i$, wh…