53 problems

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…

Duflo conjecture. The algebras

Let $bk$ be a field of characteristic zero, and let $W_n$ be the $n$-th Poisson Weyl algebra. A Poisson algebra endomorphism of $W_n$ preserves the Poisson bracket. Poisson conject…

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…

gl(N) quasi-limit conjecture. For every compact quantizable Kähler manifold, the Poisson algebra of functions is a $gl(N)$ quasi-limit.

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

Let $n\geq 2$ and let $A$ be an $n$-Poisson algebra. Full shifted Poisson complex conjecture. The $L_{\infty}$-algebra structure of the full shifted Poisson complex…

Let $G$ be a semisimple Lie group, and let ${\mathcal O}_q(G)$ and ${\mathcal O}(G)$ denote its quantized and classical coordinate rings, respectively. The topological Dixmier–Moeg…

Let $p=0$ and $n>2$. Consider the polynomial algebra $K[x_1,\ldots,x_n]$ with the Jacobian $n$-ary operation $Jac_n^S$, and, in the Poisson case, with its commutative product…

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…

Let ${\mathbf{f}}$ be a nonzero element of $\Bbbk(x,y)$, and let $K\{{\mathbf{f}}\}$ denote the associated Poisson field. Polynomial flag problem. Is there a polynomial…

Let $K$ be a Poisson field. A Poisson field has the Dixmier property if every Poisson endomorphism of $K$ is an automorphism. Dixmier problem. For which nonzero…

Let $bk$ be a field, and for $ngeq 1$ let $W_n$ be the $n$-th Poisson Weyl algebra, with $W_1=bk[x,y]$ and $fx,y g=1$. Poisson conjecture. Every nonzero Poisson algebra endomor…

Let $P=P_0\oplus P_1$ be a Poisson superalgebra, and let $\mathrm{Kan}(P)=P\oplus\bar P$ be its Kantor double, a Jordan superalgebra. The functor $\mathrm{Kan}$ sends Poisson $P$-s…

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…

Dirac reduction conjecture. The Dirac reduction with parameter $\beta=-1$ should be Poisson isomorphic to

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

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…

Invariant-integrals conjecture. The pair $(\mathfrak I,\mathfrak H_{\operatorname{tr}})$ yields a degenerate integrable system on $\Phi^{-1}(e^{\mathsf i\gamma})/{\rm U}(n)$.