20 problems

Let ${\mathfrak b}$ be a finite-dimensional basic algebra, and let ${\mathfrak n}({\mathfrak b})$ be its Lie normalizer in $P({\mathfrak b})$. Strong quantization conjecture. Every…

Let $M$ be a symplectic manifold with a finite-dimensional basic algebra ${\mathfrak b}$. Let $P({\mathfrak b})$ denote its polynomial algebra, and suppose that condition (D1),…

Let $K=K\{{\mathbf{f}}\}$ be a Poisson field, let ${^1\Gamma}_0(K)$ be the $1$-valuation gamma-cap, and let $\operatorname{fht}(K)$ denote the flag height. Gamma-cap conjecture. If…

Poisson $n$-Lie algebra construction conjecture. Then $(\mathcal A, \cdot, [-,\dots,-])$ forms a Poisson $n$-Lie algebra.

Let $H$ be a finite-dimensional Hopf algebra acting on a Weyl Poisson algebra. Rigidity conjecture. The $H$-action must factor through a group algebra. This conjecture asks whether…

Let $\g$ be one of $\W{-1}$, $W$, or $\Vir$, and let $\chi$ and $\eta$ be local functions on $\g$. Let $P(\chi)$ and $P(\eta)$ denote the associated Poisson primitive ideals, and l…

Let $\g$ be one of $\W{-1}$, $W$, or $\Vir$, and let $\chi$ and $\eta$ be one-point local functions on $\g$. Let $P(\chi)$ and $P(\eta)$ denote their associated Poisson primitive i…

Let $n\geq 2$ be an integer. Let $(L,\cdot,[\cdot,\ldots,\cdot])$ be a transposed Poisson $n$-Lie algebra, and let $D$ be a derivation of both $(L,\cdot)$ and…

Let $p=2$, let $R$ be the truncated polynomial ring from the construction, let $\mathcal L=\operatorname{Lie}(v_1,v_2)\subset\operatorname{Der}R$ be the Fibonacci Lie algebra, and…

Let $\mathfrak g$ be a simple Lie algebra, let $\mathfrak n$ be a nilradical in $\mathfrak g$, and let $U$ be the maximal unipotent subgroup associated with the chosen Borel subgro…

Let $Z$ be a Poisson algebra, and let $A$ be a quantization of $Z$. Quantization cancellation conjecture. $Z$ is Poisson cancellative if and only if $A$ is cancellative. This propo…

Let $\mathfrak g$ be a finite-dimensional Lie algebra. Its symmetric Poisson algebra is denoted by $PS(\mathfrak g)$, and its universal enveloping algebra by $U(\mathfrak g)$. Canc…

Dicksonianity conjecture. Then $\mathfrak g$ is Dicksonian.

For $n\geq 1$, let $P_n=\mathbb{K}[z_1,\ldots,z_{2n}]$ be the commutative Poisson algebra with bracket … An action is effective if its kernel is trivial, and regular if it is algeb…

Let $W_n(\mathbb{Q})$ be the $n$-th Weyl algebra over $\mathbb{Q}$ and let $P_n(\mathbb{Q})$ be the polynomial Poisson algebra in $2n$ variables with its canonical Poisson structur…

Let $W_{n,\mathbb{C}}$ be the $n$-th Weyl algebra over $\mathbb{C}$, and let $P_{n,\mathbb{C}}$ be the $n$-th commutative Poisson algebra. Kontsevich's conjecture. The automorphism…

Kontsevich–Kanel-Belov correspondence conjecture. The automorphism groups are isomorphic:

Compatibility conjecture. This diagram is commutative. The question is whether the Dunn–Lurie additivity functor is compatible with the Poisson additivity functor under the formali…

Let $V$ be a symplectic vector space over a field $\boldsymbol{F}$ of characteristic $p$, let $G$ be a finite group of order coprime to $p$ acting faithfully on $V$, and let $A$ be…

Let $\mathfrak{g}$ be a reductive Lie algebra, let $V^q$ be a finite-dimensional $U_q(\mathfrak{g})$-module, and let $V$ be its classical limit. The reduced symmetric algebra is…