26 problems

Let $R$ be a finitely generated smooth commutative algebra over $\mathbb Z$, let $P_{n,R}$ be the polynomial Poisson algebra, and let $A^{\hbar}_{n,R[\hbar]}$ be the algebra over…

Let $R$ be a finitely generated smooth commutative algebra over $\mathbb Z$, let $g\in\operatorname{Aut}(P_{n,R})$, and for each sufficiently large prime $p$ let $M_{g,p}$ be a bim…

Let $A_{n,\mathbb C}$ be the Weyl algebra of index $n$ over $\mathbb C$, and let $P_{n,\mathbb C}$ be the Poisson algebra of polynomial functions on the $2n$-dimensional affine sym…

For each $n,N\geq 1$, let ${\underline{\operatorname{Aut}}}^{\leq N}(A_{n,\mathbb Q})$ and ${\underline{\operatorname{Aut}}}^{\leq N}(P_{n,\mathbb Q})$ denote the filtered automorp…

Let $A_n$ be the complex $n$-mode Weyl algebra, and let $\mathfrak{g}$ be a finite-dimensional Lie subalgebra of $A_n$. Weyl-algebra subalgebra conjecture. Every finite-dimensional…

Sporadic Wigner–Heisenberg conjecture. The only finite-dimensional, non-nilpotent, and solvable Lie subalgebra of $\hat{A}_1$ that is not covered by the two families…

Uniqueness conjecture. The Schrödinger algebra $\mathcal{S}$ admits a unique faithful realization as a subalgebra of $\hat{A}_1$, up to automorphism. This realization is the one ex…

Let $A_n$ be the $n$th Weyl algebra, and let a finitely generated $A_n$-module be torsion when it has rank zero. A module is cyclic if it is generated by one element. Stafford's co…

Let $K$ be a field of characteristic zero. The Dixmier conjecture. Every endomorphism of the first Weyl algebra … is an automorphism. Here $A_1(K)$ is generated by $t,x$ with…

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 $\W{-1}$ 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 $K$ be a field of characteristic zero, let $k,k'\in K^n$ each have exactly $m$ nonzero elements, and let $A_n^k$ and $A_n^{k'}$ be the corresponding higher-order hom-associativ…

Generating-ideal conjecture. The ideal of all weak polynomial identities for the pair $(\mathsf{A}_1,\mathsf{V})$ is equal to $\mathcal{I}$.

Let $n$ be a positive integer, let $A_n(\mathbb{C})$ denote the Weyl algebra of index $n$ over $\mathbb{C}$, and equip $\mathbb{C}[x_1,\dots,x_n,y_1,\dots,y_{2n}]$ with its standar…

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…

Let $\Phi$ be the homomorphism from automorphisms of the Weyl algebra to polynomial symplectomorphisms constructed by reduction modulo an infinite prime, restriction to centers, in…

Let $R$ be a commutative ring, let $R_{\infty}$ be the reduction modulo the infinite prime, let $i:R\rightarrow R_{\infty}$ be the tautological inclusion, and let … be the group ho…

Let $W_n$ be the $n$-th Weyl algebra, let irreducible holonomic $\mathcal D$-modules mean irreducible holonomic modules over $W_n$, and let Lagrangian subvarieties be taken in the…

The Centralizer Conjecture over $D$. Suppose $A,B,w\in A_1(D)$ satisfy

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

Generalized Eulerian duality conjecture. The shifted module $^\sharp M^\dagger(+n)$ is a left generalized Eulerian $A_n(K)$-module.

The second $\gamma,\delta$ conjecture. For every endomorphism $f$ of $A_1$, there exist involutions $\gamma$ and $\delta$, such that $f$ is a $\gamma,\delta$-endomorphism of $A_1$.

The first $g,h$ conjecture. For every endomorphism $f$ of $A_1$, there exist $g$ and $h$, both automorphisms, both anti-automorphisms or one is an automorphism and the other is an…

The first $\gamma,\delta$ conjecture. For every endomorphism $f$ of $A_1$, there exist involutions $\gamma$ and $\delta$, each of $\gamma$ and $\delta$ is conjugate to $\alpha$, su…