58 problems

Let $n$ be a positive integer, and let the $n$-th Weyl algebra be defined over a field of characteristic zero. Dixmier's conjecture. Every endomorphism of the $n$-th Weyl algebra i…

Let $A_n(\mathbb{C})$ be the $n$-th Weyl algebra, and let $\mathbb{C}^{2n}$ carry its standard Poisson structure. A polynomial symplectomorphism of $\mathbb{C}^{2n}$ is a polynomia…

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 $bk$ 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…

Generalized Dixmier conjecture. Every endomorphism of $A_n(K)$ is an automorphism. This extends the first-Weyl-algebra problem to all Weyl algebras; the supplied text gives no reso…

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 $f(X,Y)\in\mathbb{C}[X,Y]$ be irreducible, and let $A_1=\mathbb{C}[x][\partial]$ be the first Weyl algebra. Consider pairs $(P,Q)\in A_1^2$ satisfying $f(P,Q)=0$, with the acti…

Kanel-Belov–Kontsevich conjecture. The possibly larger group ${\mathrm{Aut}}({\mathbb C}^{2n})$ is the automorphism group of the Weyl algebra ${\mathscr A}$.

Let $\mathcal D_n={\mathbb C}[x_1,\ldots,x_n,\partial_1,\ldots,\partial_n]$ be the Weyl algebra, equipped with the filtration … where $a_\alpha\in{\mathbb C}[x_1,\ldots,x_n]$. Weak…

Minimal-resolution conjecture. In the homogenized Weyl algebra, a minimal resolution exists and can be constructed by the algorithm in LaScala, since the algebra is graded; however…

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…

Non-innerness conjecture. For $t\ne 0$, $A_0(t)$ is not an inner automorphism of $W$.

Let $k$ be a field of positive characteristic, and let $A_n(k)$ be the $n$-th Weyl algebra. Let $\varphi\colon A_n(k)\to A_n(k)$ be a $k$-algebra endomorphism. Bavula's injectivity…

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 $A_n$ 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 $\mathcal{W}_n$ be the Weyl algebra over $\mathbb{C}$, and let $\mathbb{R}[x_1,\ldots,x_n;\xi_1,\ldots,\xi_n]$ be the commutative Poisson algebra with…

Let $P_k$ be the space of homogeneous polynomials of degree $k$ in the graded polynomial ring $P=\mathbb{C}[x_1,\ldots,x_n,\xi_1,\ldots,\xi_n]$, and let…

Let $P$ be a polynomial in $n$ real variables, let $\lambda\in\mathbb{C}$ satisfy $\operatorname{Re}\lambda>0$, and on an open set where $P$ is positive define the generalized func…

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 $H$ 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…