40 problems

Dixmier–Moeglin conjecture for pointed Hopf algebras. The following conditions are equivalent:

Let $\mathcal{O}$ be a finite rack of type C, let $\mathfrak{q}$ be a faithful $2$-cocycle on $\mathcal{O}$, and let $\mathscr{B}(\mathcal{O},\mathfrak{q})$ denote the associated N…

Let $\Gamma$ be an abelian finitely generated group, let $V$ be a Yetter–Drinfeld module of diagonal type over $\Gamma$, and let…

Let $A$ be a connected graded domain over $\bbk$, and write $\operatorname{GKdim}(A)$ for its Gelfand–Kirillov dimension. Artin–Stafford's gap conjecture. There is no connected gra…

Let $C$ be an associative conformal algebra. Say that $C$ is simple if it has no nonzero proper ideals, finitely generated if it is generated by finitely many elements, and of line…

Let $K$ be a field and let $A$ be a finitely generated $K$-algebra that is a domain of finite GK dimension. Let $Q(A)$ be its quotient division algebra, and let $D$ be a division s…

Let $K$ be a field and let $A$ be a finitely generated $K$-algebra that is a domain of GK dimension $d$. A chain … of division subalgebras has stratiform length $m$ when each $D_i$…

An associative algebra $A$ is an FCR-algebra if its finite-dimensional representations separate points of $A$, equivalently, if the intersection of the kernels of all finite-dimens…

Let $A=\Bbbk Q/I$ be a graded twisted Calabi–Yau algebra of dimension three with finite Gelfand–Kirillov dimension, let $M$ be the adjacency matrix of $Q$, and let $d$ be the degre…

Let $G$ be a reductive group over a $p$-adic field $F$, let $x$ be a point in its Bruhat–Tits building, and let $K_\ell:=K_{G,x,\ell}$ be the $\ell$th Moy–Prasad subgroup at $x$. W…

Let $G$ be an abelian group, and consider finite GK-dimensional pre-Nichols algebras in the Yetter–Drinfeld category $“latex {}{mathbbm{k}G}^{mathbbm{k}G}mathcal{YD} “$ over an alg…

Let $H$ be an affine noetherian pointed Hopf $\mathbb{C}$-algebra, and let $G(H)$ denote its group of grouplike elements. The pointed Dixmier–Moeglin equivalence conjecture. The fo…

Let $F$ be the $p$-adic field and let $G=\mathrm{GL}_n(F)$, with $D$ the central division algebra over $F$ of invariant $1/n$. For an admissible smooth $G$-representation $\pi$ ove…

Let $G$ be a connected reductive group over $\mathbb{Q}_p$, let $H$ be an open subgroup of $G(\mathbb{Q}_p)$, and let $d_G$ be the dimension of the flag variety of…

Let $V$ be a braided vector space of diagonal type, and let $\mathscr{B}(V)$ be its Nichols algebra. The root system of $\mathscr{B}(V)$ is the root system arising from its restric…

Let $(V,c)$ be a braided vector space of diagonal type such that … Then its generalized root system is finite. Finite GK-dimension conjecture. A Nichols algebra of diagonal type wi…

Let $W$ be the Witt algebra, let $\operatorname{U}(W)$ be its universal enveloping algebra, and let $\pi_\gamma$ be the family of representations defined in the paper, with kernels…

Let $R$ be an affine $\Bbbk$-algebra and let $W:=R(\mathbf x,\mathbf y,\sigma,h)$ be a generalized Weyl algebra of degree $m$. Write … Assume that $R$ is a finitely generated modul…

Finite-GK-dimension conjecture. One has

Finite-GK-dimension conjecture. Nichols algebras of diagonal type with finite Gelfand–Kirillov dimension have finite root system.

Angiono–Azevedo–Heckenberger conjecture. If $\mathscr{B}_{\mathfrak{q}}$ has finite Gelfand–Kirillov dimension, then its root system is finite.

Let ${\mathfrak{q}}$ be a braiding matrix of diagonal type, let ${\mathcal B}_{\mathfrak{q}}$ be its Nichols algebra, and let $\widetilde{\mathcal B}_{\mathfrak{q}}$ be the disting…

Let $\mathbb K$ be a field of characteristic zero, and let $W_+$ be the positive Witt algebra with basis $\{e_n:n\geqslant 1\}$ and Lie bracket $[e_i,e_j]=(j-i)e_{i+j}$. Its univer…

GK-dimension conjecture. One has

Let $V$ be a finite-dimensional braided vector space of diagonal type, and let $\mathscr{B}(V)$ denote its Nichols algebra. An arithmetic root system conjecture. If … then $V$ has…