40 problems

Let $\mathfrak{g}$ be a finite-dimensional $k$-step nilpotent Lie algebra. Nilpotent minimal-graph-admissibility conjecture. If $k\leq 2$, then $\mathfrak{g}$ is minimal-graph-admi…

Let $D$ be a rank $2$ distribution, let $\aleph_D=\{\lambda\in\mathcal R_D:{\cal A}_\lambda\neq 0\}$, and let $g_{r,2}$ denote the free nilpotent $r$-step Lie algebra with two gene…

A Lie algebra is characteristically nilpotent when every derivation of it is nilpotent. Kharitonov's conjecture. Every nilpotent Lie algebra which does not admit an affine structur…

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…

Deré's conjecture. If $L$ is isomorphic to its complex conjugate, then $L$ is defined over $\mathbb{R}$; equivalently, there exists a real Lie algebra $L_{\mathbb{R}}$ such that

Let $D$ be a distribution with an EKR presentation, and consider the nilpotency orders associated with its EKR normal forms. Mormul's conjecture. For a given distribution $D$, the…

Let $\mathfrak{g}$ be a finite-dimensional nilpotent Lie algebra over $\mathbb{k}$, and set $A=\operatorname{CE}(\mathfrak{g})$. Let $S$ be the symmetric algebra governing the Kosz…

Toral rank conjecture. One has

Let $(\mathfrak{g}, \{J_{\alpha}\})$ be a nilpotent hypercomplex Lie algebra. Here, $\mathbb{H}$-solvability means that the descending $\mathbb{H}$-central series defined by…

Flatness conjecture. Every two-step nilpotent hypersymplectic Lie algebra is flat.

Let $n$ and $p$ be integers with $n<p$, and let $L$ be an $n$-Engel Lie algebra over $\mathbb{F}_p$. For every $x e L$, let $I(x)$ be the ideal generated by $x$; call $L$ **$k$-str…

Let $0\ne a\in\mathbf L$ be written in the basis as $a=\sum_{j=n}^{m}r_{j-3}v_j$, with $r_j\in R$. The preceding result gives the bound $a^{2^{m-n+2}}=0$. Optimality conjecture. Th…

Conant-independence conjecture. Over models, Conant-independence coincides with \mathop{\mathpalette\setbox0=\text{x}\kern\wd0\text to 0pt{\hssmidhss} lower.9ht0 to 0pt{hss\s…

Let $\mathcal{N}$ be a nilpotent Lie algebra that is not characteristically nilpotent, and assume that all nil-independent derivations are represented by upper-triangular matrices.…

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 (…

Let $\mathfrak{n}$ be a complex nilpotent Lie algebra that is not characteristically nilpotent. Let $\mathfrak{s}$ and $\tilde{\mathfrak{s}}$ be solvable Lie algebras with nilradic…

Let $\mathfrak g$ be a finite-dimensional nilpotent Lie algebra. Write $Z(\mathfrak g)$ for the center of its universal enveloping algebra, and let the algebra of polynomial invari…

Let $\ngo$ be a nilpotent Lie algebra, and let $\Cone$ be a cone in the maximal torus of derivations of $\ngo$. Let $\sg$ be a solvable Lie algebra with nilradical $\ngo$. The nota…

Let $\ngo$ be a nilpotent Lie algebra. Consider the space of diagonalizable derivations of $\ngo$ for which the corresponding one-dimensional solvable extension admits a left-invar…

Let $L$ be a finite-dimensional nilpotent Lie algebra defined over the rational numbers, and let $C(L)$ denote its center. Nilpotent Lie algebra toral rank conjecture. Then … This…

Rational-form invariance conjecture. If $\mathfrak{n}_{\mathcal{G}}^\mathbb{Q}$ admits an Anosov automorphism, then every rational form $\mathfrak{m}^\mathbb{Q}$ of…

Let $\mathfrak{n}$ be a nilpotent Lie algebra with center $\mathfrak{z}$. Toral rank conjecture. The inequality … continues to hold in the k-ary setting for free 2-step nilpotent k…

Let $\mathfrak{n}$ be a nilpotent Lie algebra with center $\mathfrak{z}$. Toral rank conjecture. The total homology satisfies … The conjecture concerns a lower bound for the total…

Let $\mathfrak{n}$ be a nilpotent Lie algebra, let $\mathfrak{t}(\mathfrak{n})$ be a maximal torus of diagonalizable derivations, let $\mathfrak{t}(\mathfrak{n})_{\mathrm{srn}}$ be…

Let $\mathfrak{n}$ be a nilpotent Lie algebra, let $\mathfrak{t}(\mathfrak{n})$ be a maximal torus of diagonalizable derivations, and let…