17 problems

Goodearl's conjecture. If $R$ is an affine algebra of GK-dimension $2$, then its Jacobson radical $J(R)$ is nil.

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 $D_i$…

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…

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 $H$ 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 $V$ 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 $W$ 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…

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

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

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…

Let ${\rm U}(W_+)$ be the universal enveloping algebra of the positive Witt algebra $W_+$ over a field of characteristic zero. The algebra ${\rm U}(W_+)$ has just infinite Gelfand–…

Leading-coefficient ratio conjecture. There exists a constant $C$, depending only on the Lie algebra $\mathfrak{g}$ and $d$, such that for all $M\in\mathcal{O}_\lambda^d$,

Let $G$ be a finite abelian group and let $A$ be a $G$-graded PI-algebra. For each $k\geq 1$, write $\operatorname{GKdim}_k^G(A)$ for the $k$-th $\mathbb{Z}_2$-graded Gelfand–Kiril…

Let $k$ be a finite field and let $A$ be a finitely generated $k$-algebra of quadratic growth. Finite-field quadratic-growth conjecture. Then $A$ is not simple. The conjecture is m…