23 problems

Bahturin–Regev conjecture. The following statements hold:

Let $\mathcal{A}_{k+1}$ be the algebra generated by the generators and relations of Definition of the paper, and let $E_{k+1}^{(q,\alpha_1)}$ be the corresponding element. Speciali…

Let the maps in Proposition and Theorem be the natural maps associated with the $\mathbb Z_3$-symmetric down-up algebra $\mathbb A=\mathbb A(\alpha,\beta,\gamma)$. Injectivity conj…

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…

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

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 $U(\mathfrak{so}(n))$ 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…

Let $A$ and $H$ be as in Theorem textup{(c)}. For $J\subset J'$ in $H\text{-}\operatorname{Spec} A$, let $\phi^p_{JJ'}$ and $\phi^s_{JJ'}$ be the maps between closed subsets of the…

Assume that $\operatorname{char} k=0$. Let $A$ and $A'$ be suitably generic members of some flat family of quantized coordinate rings for an affine variety $V$ with semiclassical l…

Let $A$ be a quantized coordinate ring of an affine variety $V$. Topological quotient conjecture. The spaces $\operatorname{Spec} A$ and $\operatorname{Prim} A$ are compatible topo…

Master pentagon identity. The generating functions satisfy

Let $A:=\Bbbk_{-1}[\mathbf{x}]$ and let $G:=C_n$ act as in the paper. A graded isolated singularity means that the invariant ring has the isolated-singularity property in the grade…

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.

Let $A,B$ be a tridiagonal pair over $\mathbb F$ having $q$-Racah type in the sense of Bockting-Conrad, and let $V$ be its underlying vector space. Let $\square_q$ denote the quant…

Let $V$ be a finite-dimensional irreducible $\widetilde \square_q$-module, where $\widetilde \square_q$ and $\boxtimes_q$ are the algebras appearing in the paper, with generators…

Generic-basis recurrence. With these notations, one has

General block-size braid-group action conjecture. The braid-group action for general $m\times m$ matrix blocks should have the same form as and, with

Let $D$, $V$, and $V'$ be as above, and let $K_1\cdot K_2\cdot\ldots\cdot K_m$ be a convolution complex in … where each $K_a$ is either $\mathscr E^{(n),\Omega}_{\bullet}$ or…

Power-of-two Pfaffian conjecture. The absolute value of the Pfaffian satisfies

Primitive-prime enumeration conjecture. The number of primitive $\mathcal{H}$-primes in ${\mathcal O}_q(M_{3,n})$ is