23 problems

Let $\mathcal{L}$ be a finite dimensional restricted Lie algebra over an algebraically closed field $\mathfrak{k}$ of characteristic $p>0$, and let $\mathcal{N}(\mathcal{L})$ denot…

Kostrikin–Šafarevič conjecture. $L$ is of classical or Cartan type.

Let $G_\mathbb{F}(p)$ be the maximal pro-$p$ quotient of the absolute Galois group $G_\mathbb{F}$ of a field $\mathbb{F}$, and let the Jennings–Zassenhaus filtration of…

Let $\mathfrak{g}=\mathfrak{gl}(m\cdot\1+nP)$, or let $\mathfrak{g}$ be any restricted subalgebra of it, and let $X\in\mathfrak{g}$ with $X\notin\ker d$. Write $X^{[2]}$ for the re…

Property PB conjecture. All restricted Lie algebras satisfy Property PB regardless of representation type.

Property PC conjecture. For any restricted Lie algebra ${\mathfrak g}$ over $k$, ${\mathfrak g}$ satisfies Property PC if and only if ${\mathfrak g}$ is of tame or finite represent…

Let $p=2$, let $\mathcal L=\operatorname{Lie}(v_1,v_2)$ be the Fibonacci Lie algebra, and let $\mathbf L=\operatorname{Lie}_p(v_1,v_2)$ be its Fibonacci restricted version. The sou…

Let $p=2$ and let $\mathbf L=\operatorname{Lie}_p(v_1,v_2)$ be the Fibonacci restricted Lie algebra. Infinite-presentation conjecture. The algebra $\mathbf L$ is infinitely present…

Let $p=2$, let $R$ be the truncated polynomial ring from the construction, let $\mathcal L=\operatorname{Lie}(v_1,v_2)\subset\operatorname{Der}R$ be the Fibonacci Lie algebra, and…

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…

Let the restricted Fibonacci Lie algebra be the restricted Lie algebra associated with the Fibonacci Lie algebra studied in the paper. A presentation is infinitely presented if no…

Kac's conjecture (KW2). The integer

Kac–Weisfeiler conjecture (KW1).

Let $\mathcal{L}$ be a finite dimensional restricted Lie algebra over an algebraically closed field $\mathfrak{k}$ of characteristic $p>0$, and let $V$ be an irreducible restricted…

Maximal-torus bound. There are at most $p^{{\rm mt}\text{-}\!\operatorname{rank}(\mathfrak{g})}$ nonisomorphic simple $\mathfrak{u}(\mathfrak{g},\chi)$-modules.

Let $(\mathfrak{g},[p])$ be a restricted Lie algebra with connected automorphism group $G:=\operatorname{Aut}_p(\mathfrak{g})^{\circ}$, and let $\mathfrak{t}\subseteq\mathfrak{g}$…

Let $k$ be an algebraically closed field of characteristic $p$, and let $\mathfrak{g}$ be a restricted Lie algebra. Its index is … where $\mathfrak{g}^\chi$ is the coadjoint stabil…

Restricted cohomological-dimension-one structure conjecture. Any Lie $p$-algebra $L$ with $\operatorname{cd}_*(L)=1$ is of the form

Two-dimensional subalgebra conjecture. $L$ contains a two-dimensional subalgebra.

Universal upper-bound conjecture. The canonical upper bound for the dimension of the projective cover of the one-dimensional trivial module holds for every finite-dimensional restr…

Let $k$ be the underlying field, let $S_n$ be the special Lie algebra of divergence-zero derivations in $n$ variables, let $W_{n-1}$ be the Witt algebra in $n-1$ variables, and let…

Premet's conjecture. There exists a nonempty Zariski open subset $V\subseteq{\mathfrak g}$ consisting of regular elements such that, for any $u,v\in V$, the Cartan subalgebras

Let $\mathscr L$ be an $n$-dimensional restricted Lie algebra, and let $r$ be the minimal dimension of the centralizers of all $\chi\in\mathscr L^*$. Write $M(\mathscr L)$ for the…