18 problems

Let $\g$ be one of $\W{-1}$, $W$, or $\Vir$, and let $\Ua(\g)$ be its enveloping algebra. ACC conjecture. The algebra $\Ua(\g)$ satisfies the ascending chain condition on primitive…

Let $\g$ be one of $\W{-1}$, $W$, or $\Vir$, and let $\Ua(\g)$ be its enveloping algebra. Complete-primality conjecture. Every primitive ideal of $\Ua(\g)$ is completely prime. The…

Let $\W{-1}$ denote the negative Witt algebra, let $\Ua(\W{-1})$ be its enveloping algebra, and let $A_k$ denote the relevant $k$-th Weyl-algebra target. For each $n$, write…

Kac–Weisfeiler conjecture (KW1).

Let $L$ be the coefficient field, let $\mathfrak{h}$ be a semisimple $L$-Lie algebra, let $\mathfrak{h}_0\subset\mathfrak{h}$ be an $\mathcal{O}$-Lie lattice, and let…

Let $\mathfrak{g}$ be the Lie algebra of a linear algebraic group over $\mathbb{C}$. For $r,s\in\mathbb{Z}_+$, let $\mathcal{D}_{r,s}=\mathcal{D}_r\otimes\mathbb{C}[y_1,\dots,y_s]$…

Let $V\!ir$ be the Virasoro algebra, let $z$ denote its central element, and let $\operatorname{U}(V\!ir)$ be its universal enveloping algebra. The Virasoro central-quotient simpli…

Let $\mathfrak{g}$ and $\mathfrak{g'}$ be finite-dimensional Lie algebras over $\mathbb{C}$. Write $U(\mathfrak{g})$ for the enveloping algebra of $\mathfrak{g}$, and likewise for…

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…

Higher baby Verma quotient conjecture. Every irreducible $U_\chi^{[r]}(G)$-module is a homomorphic image of $Z_\chi^{[r]}(N,\lambda_r)$ for some irreducible…

Let $W_+$ be the positive Witt algebra over a field of characteristic zero, and let ${\rm U}(W_+)$ be its universal enveloping algebra. The two-sided ideals of ${\rm U}(W_+)$ satis…

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–…

Let $\mathfrak{g}_1$ and $\mathfrak{g}_2$ be finite-dimensional Lie algebras over $\mathbb{C}$, and let $\mathfrak{U}\mathfrak{g}_1$ and $\mathfrak{U}\mathfrak{g}_2$ denote their e…

Decomposition conjecture. Assume $\lambda$ is strongly typical and regular, and $V\cong\Pi V$. Then

Bimodule direct-summand conjecture. Assume $\lambda$ is strongly typical and regular. Then the bimodule $U/I$ is a direct summand of

Injectivity conjecture. Assume $\lambda$ is strongly typical and regular. Then $(U/I)^{\mathrm{ad}}$ is a direct sum of injective finite-dimensional modules.

Assume $p=2$ and $n\ge4$. For a sequence $i=(i_1,\dots,i_l)$, let $a_{i,k}=a_{i_1,k}\dots a_{i_l,k}$; write $i..j$ for the consecutive sequence from $i$ to $j$, and let $L_l[m]$ be…

Let $p$ be the characteristic of the field $\mathbb{K}$, let $X=\{e_{i,i+1}^{(p^k)}\mid 1\le i\le n-1,\\ k\in\mathbb{N}_0\}$, and write $a_{ik}=e_{i,i+1}^{(p^k)}$. Equip $X$ with t…