250 problems

Let $\mathcal L$ be the Fibonacci Lie algebra, let $\lambda=(1+\sqrt 5)/2$, and let $\mathbf E(\mathcal L,t)$ denote its Euler characteristic. Euler-characteristic asymptotic conje…

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…

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…

Let $A$ be a simple locally finite associative algebra over $\mathbb{F}$, let $L=[A,A]$, and let $B$ be a Jordan-Lie inner ideal of $L$. Regularity conjecture. Every Jordan-Lie inn…

Let $p>2$ and let a finite-dimensional Lie algebra or Lie superalgebra have an indecomposable Cartan matrix normalized as in the source's normalization convention. A Chevalley basi…

Quadratic growth conjecture.

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

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

Basis conjecture. One has

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…

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.

Projective-module Loewy-length conjecture. If $\chi$ has standard Levi form with $I=\Pi$ and $\lambda$ is $p$-regular, then

Let $\mathfrak{g}$ be a simple Lie algebra over $\mathbb{C}$, fix a Cartan subalgebra $\mathfrak{h}$, and let $\Phi$ be the associated root system with Weyl group $W$. Choose posit…

Let $M\subset\mathbb C^{n+k}$ be a totally nondegenerate CR model of length $\rho\geq 3$, with graded infinitesimal CR automorphism algebra … Here, rigidity means that the positive…

Pollack's lifting conjecture. There exists a linear combination $T$ of brackets of $\varepsilon_{2i}$, containing at least three $\varepsilon_{2i}$ with $i>1$, such that

Let ${\bf k}={\mathbb R}$ or ${\mathbb C}$, let $\G$ be a finite-dimensional ${\bf k}$-Lie algebra, and let $\hat{\mathfrak f}_{2}$ denote the completed free Lie algebra on $x,y$.…

Let $\overline{\mathfrak{g}}$ be a finite-dimensional simple Lie algebra, and let $\lambda\in \overline{P}_+\backslash\{0\}$ be a non-zero dominant integral weight. The relative as…

Let $\mathfrak{h}$ be the Cartan subalgebra and let $\Lambda$ denote the set of parameters for which the corresponding quantization exists, viewed as a semigroup under the semigrou…

Let $\mathrm{Dif}_1$ denote the Lie algebra of differential operators of order at most one on the circle, and let $\Psi_3,\widetilde\Psi_5,\widetilde\Psi_7,\widetilde\Psi_9,\dots$…

Let ${\frak g}$ be a finite-dimensional Lie algebra with a metric and orthonormal basis $\{{\frak g}_a\}_{a=1}^{\dim{\frak g}}$, and let $R$ be a finite-dimensional representation.…

Let ${\cal SW}(V)$ be the Schur–Weyl category, let $V_{\lambda}$ denote its distinguished objects, and let ${\bf g}=sl(V_S)$ act on these objects. A linear map between two objects…

Let $(\Gamma_1,\Gamma_2,\tau)$ be an admissible triple and let $r^0\in\mathfrak h\otimes\mathfrak h$ satisfy the equations referenced in the source. Define $\tilde r^0=r^0-P^0/2$,…

Let $\lambda\in\mathbb C$, let ${\mathfrak{gl}}(\lambda)$ be embedded in $\mathrm{Dif}_1$, and let … be the first page of the associated Feigin Serre–Hochschild spectral sequence.…

Let $\mathfrak{g}$ be a semisimple Lie algebra over an algebraically closed field of characteristic $0$, and let $x\in\mathfrak{g}$. By $C_{\mathfrak{g}}(x)$ denote the centralizer…

Let $\mathfrak{a}$ be a simple Lie algebra, let ${\mathbf d}\geq 1$, and let $\psi_{\mathbf d}$ be the real-valued polynomial obtained by restricting the potential $\Psi$ to the sp…