68 problems

Let $X$ be a variety over a perfect field $k$. Write $\operatorname{Der}_{\mathbb{C}}(\mathcal{O}_X)$ for the module of derivations of $\mathcal{O}_X$ over $\mathbb{C}$. Zariski–Li…

Divisibility conjecture. For any $m,i\geq 0$, the polynomial $\widetilde{f^m_i}(x,y)+\widetilde{f^m_i}(y,x)$ is divisible by

Let $\mathcal{A}$ be a commutative Banach algebra, and let $D:\mathcal{A}\to\mathcal{A}$ be a derivation, meaning that $D(ab)=aD(b)+bD(a)$ for all $a,b\in\mathcal{A}$. Singer–Werme…

The LFED Conjecture. Images of locally finite $K$-derivations and $K$-$\mathcal{E}$-derivations of $K$-algebras are Mathieu subspaces.

Yan's conjecture. If $d$ is a simple derivation of $R_n$, then $\operatorname{Aut}(R_n)_d$ is conjugate in $\operatorname{Aut}(R_n)$ to a subgroup of the group of translations.

Automorphism conjecture. Every automorphism of the Lie algebra $W_{\infty}(\mathbb{F})$ is a conjugation by an automorphism…

Let $K$ be a field of characteristic zero, ${\mathcal A}$ a commutative $K$-algebra, and let $\delta$ be a locally nilpotent (LN) $K$-derivation or a locally nilpotent $K$-…

Hussain–Ma–Yau–Zuo's conjecture. The graded algebra

Let $A$ be an integral domain that is a finitely generated algebra over a field $k$ of characteristic $0$. Write $\operatorname{Der}(A)$ for the $A$-module of derivations of $A$. Z…

Let $R$ be a Noetherian local ring with residue field $k$, and let $\operatorname{Der}_k(R)$ denote the $R$-module of $k$-derivations of $R$. Herzog–Vasconcelos conjecture. If … th…

Let $A$ be a normal graded algebra whose associated graded singularity is isolated, and consider the homogeneous $k$-linear derivations of $A$ with respect to a positive grading. W…

Conjecture. For any derivation $D$ the algebra $k[X]^D$ is generated by Casimir elements.

Morris's conjecture. Whenever $R(X)$ has no non-zero, bounded point derivations, $R(X)$ must be weakly amenable.

Let $\operatorname{Mat}_n$ be the algebra of $n\times n$ matrices, and let $D$ be a quasiderivation of this algebra. Višik's quasiderivation conjecture. Every quasiderivation has t…

Let $\frak{g}$ be a characteristically nilpotent Lie algebra of derivations. An outer derivation is a derivation not belonging to the space $IDer(\frak{g})$ of inner derivations. T…

Vukman–Kosi-Ulbl conjecture. The map $d$ must be a derivation whose image is contained in the center $Z(R)$.

Let $L=L_\mathbf{0}\oplus L_\mathbf{1}$ be a quasireductive Lie superalgebra, and let an almost inner derivation of $L$ be a derivation that is almost inner, while an inner derivat…

Yau's conjecture. The Tjurina algebra has no negative weighted derivations. This conjecture is still open to the authors' knowledge, although many cases have been resolved.

Let $k$ be a field of characteristic zero and let $A$ be a $k$-algebra of finite type. For each integer $q\ge 1$, let $Der^q(A)$ denote the module of differential operators of orde…

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…

Baltazar's finiteness conjecture. The isotropy group of $d$ is finite.

Let $p$ be an odd prime, let $K=\mathbb{Q}(\zeta)$ with $O_K=\mathbb{Z}[\zeta]$, and let $A$ be the matrix defined by … for $\beta=\sum_{i=0}^{p-2}b_i\zeta^i\in K$. Let $D$ be a…

The determinant-$p$ conjecture. The matrix $A=[a_{ij}]$ is a $(p-1)\times(p-1)$ matrix with determinant $p$.

Let $R$ be a Noetherian local ring with residue field $k$, and let $\operatorname{Der}_k(R)$ denote the $R$-module of $k$-derivations of $R$. Strong Zariski–Lipman conjecture. If ……

Consider a nilpotent Lie algebra with a maximal torus and root subspaces, and suppose that all nil-independent derivations are represented by upper-triangular matrices. Condition (…