595 problems

Let ${\mathfrak q}$ be a reductive Lie algebra and let $\xi\in{\mathfrak q}^*$. The index ${\rm ind}\,{\mathfrak q}^\xi$ is the index of the stabiliser of $\xi$ under the coadjoint…

Vergne's conjecture. There are no rigid complex nilpotent Lie algebras in $\mathrm{L}_n$.

Gelfand–Kirillov conjecture. The skew field of fractions of $U$ should be a suitable Weyl field.

Pour $n\geq 1$, soit $\iota^\bullet(n):\mathcal{B}^\bullet(F,n)\rightarrow\mathrm{CE}^\bullet(\mathcal{C}(F))_n$ le morphisme de complexes induced by motivic polylogarithms. Conjec…

Kashiwara–Vergne conjecture. The specified set of Kashiwara–Vergne equations has a solution in the group of tangential automorphisms of $\widehat{\mathfrak{f}}_2$.

Grunewald–O'Halloran conjecture. Every complex finite-dimensional nilpotent Lie algebra of dimension greater than two is the degeneration of another Lie algebra.

Let … =^{-}oplusoplus^{+}. … , and let the diagram automorphism group of $^{+}$ have order $1$, $2$, or $3$. Andruskiewitsch–Dumas conjecture. The automorphism group of $U_q(^{+})$…

Basis conjecture. One has

First Kac-Weisfeiler conjecture. The equality

Deligne–Drinfeld conjecture. $\mathfrak{grt}_1$ is the completed free Lie algebra with generators $\sigma_3,\sigma_5,\sigma_7,\dots$ of homogeneous weights $3,5,7,\dots$.

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…

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

Grunewald–O'Halloran conjecture. For every $n$-dimensional nilpotent Lie algebra $\mathcal A$, there exists an $n$-dimensional non-nilpotent Lie algebra $\mathcal B$ such that

Let $\Kee$ be an algebraically closed field of characteristic $p>7$, and consider simple finite-dimensional restricted Lie algebras over $\Kee$. Kostrikin–Shafarevich conjecture. T…

Winnie-the-Pooh conjecture. The Lie algebra $sl(n)$ has an orthogonal decomposition if and only if

Let $\mathfrak{grt}_1$ be the Grothendieck–Teichmüller Lie algebra, and let $\widehat{\operatorname{Lie}}(\sigma_3,\sigma_5,\sigma_7,\ldots)$ denote the free complete Lie algebra o…

Let $\Lambda$ be the algebra of universal Lie-theoretic diagrams introduced in the paper, and let $t$ and $x_n$ denote its special diagrams. Vogel's generation conjecture. The alge…

Let $\mathfrak g=\bigoplus_{m=1}^{\infty}\operatorname{gr}^m\mathfrak g$ be the graded ${\mathbb Z}_p$-Lie algebra arising from the weight filtration on the relevant pro-$p$ Galois…

Let $K$ be a field of characteristic zero, let $\mathfrak{g}$ be a Lie algebra over $K$, and let $U(\mathfrak{g})$ denote its universal enveloping algebra. Sierra–Walton conjecture…

Drinfeld's conjecture. The inclusion is surjective; equivalently, $\mathfrak{grt}$ is isomorphic to the free graded Lie algebra with one generator in every odd degree at least $3$.…

Mukhin–Varchenko conjecture. Under these representation-theoretic assumptions on $\alpha_1$ and $\alpha_2$, the corresponding $\mathfrak g$-Selberg integral can be expressed as a r…

Let $\mathfrak{g}$ be a Lie algebra, and let $\left\{\,, \,\right\}$ denote its Lie–Poisson bracket. A family of functions on $\mathfrak{g}^*$ is complete in the usual integrabilit…

Quasi-reductivity conjecture. An index-one Lie poset algebra of type A, B, C, or D is contact if and only if it is quasi-reductive.

Let $\mathfrak{g}$ and $\mathfrak{n}$ be Lie algebras, with $\mathfrak{g}$ reductive and $\mathfrak{n}$ semisimple. Suppose that the pair $(\mathfrak{g},\mathfrak{n})$ admits a pos…

Coll et al.'s nonexistence conjecture. No general linear greatest common divisor formula exists for the index of seaweed algebras when there are more than three parts in the top co…