80 problems

For a positive integer $N$, let $X_j^{(N)}(z)$, for $j=0,\dots,N$, be the tetrahedral $X$-operators depending on the spectral parameter $z$. Commutativity conjecture. For an arbitr…

Let $B$ be a quantized coordinate ring and let $\overline{B}$ be its corresponding semi-classical limit, with the induced Poisson structure. Assume that the base field is algebraic…

A quantized coordinate ring is a noncommutative algebra serving as a quantum analogue of the coordinate ring of an affine algebraic variety, and its prime spectrum consists of its…

Let $G_{n}^{\,\prime}(q)$ be the affine Sergeev algebra, and let $\psi_{\Lambda^{r}}T_{w_{\Lambda^{r}}}^{\,-1}\in G_{n}^{\,\prime}(q)$ be the element introduced as the analogue of…

Let $W^{(j)}(z)$ be the currents of the deformed $W_n$ algebra (DWA), and let ${\overline X}_{\alpha}(\lambda)$ denote the screening operators defined by (SET1). An intertwiner is…

Let ${\cal U}(osp(1|2n))$ be the enveloping algebra of $osp(1|2n)$, equipped with its usual adjoint action and with the action $\triangleright$ considered in the paper. Regard it a…

Equivalence conjecture for truncated moduli algebras. Looking at moduli algebras corresponding to truncated quantum symmetries should be equivalent to dealing with truncated moduli…

Moduli algebra conjecture. The moduli algebra is a quantum algebra of observables of Hamiltonian Chern–Simons theory.

First-Casimir conjecture. For every choice of the real constants $a_k$, there exist real constants $b_k$ such that

Let $A$ be an associative algebra with generators $x_1,\ldots,x_n$ and defining relations … where $p_{ij}\in\mathbb{C}[x_1,\ldots,x_{i-1}]$ if $i>j$, $(h_{ij})$ is an antisymmetric…

Let $V$ be a variety with commutative coordinate ring $\mathcal{O}(V)$, and let $\mathcal{O}_q(V)$ be a more general quantization of it. Goodearl's conjecture. The space…

Assume $\mathbb K$ is algebraically closed. Let $V$ be a finite-dimensional vector space over $\mathbb K$ of positive dimension, and let $A,A^*$ be a tridiagonal pair on $V$. Let…

Assume $K$ is algebraically closed, $q\in\mathbb K$ is not a root of unity, $V$ is a finite-dimensional nonzero $K$-vector space, and $R,L:V\to V$ are nilpotent linear maps satisfy…

Uniqueness conjecture. In $U_q^w$ and $U_q^v$, the only zero divisors annihilating all generators are $J_w-1$ and $J_v-1$, respectively.

A $1$-$1$ tangle invariant is an invariant assigned to a tangle with one incoming and one outgoing endpoint; reversing a knot reverses its orientation. Conjecture. The $1$-$1$ tang…

Quantum observable normal-form conjecture. Every quantum observable ${\cal O}$ can be presented as

Boundary conjecture. Closed conformal field theory is the boundary of the set of open conformal field theories.

Let $A_n(1,p_{ij})$ be the quantum Weyl algebra over a field $K$ with parameter $q=1$, and suppose that $\operatorname{char}(K)=0$. Its Krull dimension is known to be $n$. Global d…

Feigin's conjecture. The map $\Phi(i_1,\ldots,i_N)$ is an embedding, and, for at least one special reduced decomposition of $w_0$, it extends to an isomorphism

Let $U_q$ be the quantum group considered in the paper, and let $\lambda_1,\ldots,\lambda_{n-1}$ be the labels of the highest-weight vector (HWV) and lowest-weight vector (LWV). Fo…

Let $\mathcal J$ and $\hat{\mathcal J}$ be the ideals defining the quotient algebras $\mathcal A_{\mathcal J}$ and $\mathcal A_{\hat{\mathcal J}}$, and let $\partial_{t_\ell}$,…

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…

Conjecture 1. The same mutual commutativity relation as in the $A_l^{(1)}$ and $D_l^{(1)}$ cases holds for ${\mathfrak g}=E_l^{(1)}$:

Let $U(\mathfrak{so}(n))$ be the universal enveloping algebra, and let $\mathfrak{O}_{\overline{0}}$ denote the even subalgebra of the TAMA. For a sequence $A=(a_1,a_2,a_3,a_4)$ of…