30 problems

Gelfand–Kirillov conjecture. The skew field of fractions of $U$ should be a suitable Weyl field.

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…

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 $\mathfrak g(\infty)$ be a locally simple Lie algebra, and let $\operatorname{U}(\mathfrak g(\infty))$ be its universal enveloping algebra. An augmentation ideal is an ideal ar…

Gel'fand–Kirillov conjecture. The quotient rings satisfy

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 $U(L_{\alpha,_i})$ be the enveloping algebra of the Takiff algebra $L_{\alpha,_i}$. Let $\mathfrak{I}(\mu)$ be the primitive ideals described in Theorem 4.3 for…

Let $G$ be a classical algebraic group with Lie algebra ${\mathfrak g}$, and let $U({\mathfrak g})$ be its enveloping algebra. Losev's orbit method gives a natural embedding from t…

Kac's conjecture (KW2). The integer

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 $W_+=t^2\Bbbk[t]\frac{d}{dt}$ be the positive Witt algebra, and let $U(W_+)$ denote its enveloping algebra. Ascending chain condition conjecture. The algebra $U(W_+)$ satisfies…

Maximal-torus bound. There are at most $p^{{\rm mt}\text{-}\!\operatorname{rank}(\mathfrak{g})}$ nonisomorphic simple $\mathfrak{u}(\mathfrak{g},\chi)$-modules.

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…

Weight-extension conjecture. Each extension of $\lambda_{r-1}$ to $\lambda_r\in\Lambda_\chi^r$ determines an irreducible $\widehat{U_\chi^{[r]}(B)}$-module structure on the…

Higher baby Verma module conjecture. The higher baby Verma modules should provide the correct analogue of baby Verma modules for the higher universal enveloping algebras.

Let $W_+$ be the positive Witt algebra and let $f$ be a nonzero element of ${\rm U}(W_+)$. Write $o(f)$ for the order used in the paper. The finite-GK-dimension conclusion of the q…

Let $W$ be the Witt algebra and $V$ the Virasoro algebra, with ${\rm U}(W)$ and ${\rm U}(V)$ their universal enveloping algebras. The analogues for ${\rm U}(W)$ and ${\rm U}(V)$ of…

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…

Let $\mathfrak g(\infty)$ be one of the locally simple Lie algebras $\mathfrak{sl}(\infty)$, $\mathfrak o(\infty)$, or $\mathfrak{sp}(\infty)$. Consider its universal enveloping al…