185 problems

In 1977, Kac classified the finite-dimensional Lie superalgebras, while Cartan's list of infinite-dimensional Lie algebras is $(W_n,S_n,H_n,K_n)$. Kac's conjecture. The infinite-di…

Boundary-level rationality conjecture. The vertex operator superalgebra $L_{\widehat{\mathfrak{g}}}(\mathcal{k},0)$ is rational in the category $\mathcal{O}$, and its irreducible w…

Let $\mathrm{L}=\bigoplus_{i\in\mathbb{Z}}\mathrm{L}_i$ be a simple $\mathbb{Z}$-graded Lie superalgebra of growth one whose even part contains the Witt algebra $\mathrm{Vir}$. Suc…

Let $\mathfrak{g}$ be an arbitrary quasi-reductive Lie superalgebra and let $\mathfrak p$ be a parabolic subalgebra of $\mathfrak{g}$. Write $\mathcal{PI}^{\mathfrak p}$ for the fu…

Let $\mathfrak{gl}_{m|n}$ be the general linear Lie superalgebra, and let ${\rm Y}(\mathfrak{gl}_{m|n})$ be its Yangian. The formal power series $b_{m|n}^-(u)$, called the quantum…

Let $\frak{g}$ be a basic classical Lie superalgebra, and consider the Duflo–Serganova functors $DS_x$ on $\frak{g}$-modules. A module is semisimple if it is a direct sum of simple…

Let ${\mathfrak g}$ be a basic simple Lie superalgebra, let $\lambda$ be a weight, and let $L(\lambda)$ be the corresponding finite-dimensional irreducible representation. Write…

Generalized Kac–Wakimoto conjecture. Every simple ${\mathfrak{g}}$-supermodule $J$ is ambidextrous. If $L$ is another simple supermodule satisfying

Let $\mathfrak{g}$ be a classical Lie superalgebra, let $\mathfrak{e}$ be its detecting subalgebra, and let $L$ be a simple $\mathfrak{g}$-supermodule. Denote by…

Let ${\cal U}(osp(1|2n))$ be the enveloping algebra of $osp(1|2n)$, equipped with its usual adjoint action and with the action $\triangleright$ considered in the paper. Regard it a…

The nqc-type conjecture. The composition factors of $\overline{V}(\Lambda)$ are in one-to-one correspondence with the codes $\Sigma^c$ determined from $nqc(\Lambda)$, with the code…

Hughes' permissible-code conjecture. There is a bijection between the composition factors of $\overline{V}(\Lambda)$ and the permissible codes.

Let $\mu$ be integral dominant and $r$-fold atypical with respect to the roots $\gamma_1<\gamma_2<\cdots<\gamma_r$ ($\gamma_i\in\Delta_{1,+}$). Let $\lambda_\theta$, for…

The composition-factor multiplicity conjecture. For each of the $2^r$ integral dominant weights $\lambda_\theta$, $a_{\lambda_\theta,\mu}=1$, and $a_{\lambda,\mu}=0$ for every othe…

Let $J$ be a finite subset of the real indices, and let ${\frak g}_{0}^{(J)}$ and ${\frak g}_{-}^{(J)}$ be the subalgebras in the triangular decomposition associated with $J$. For…

For a Lie superalgebra with Cartan matrix generated by Chevalley generators, let non-Serre-type relations mean defining relations beyond the Serre-type relations. Non-Serre encodin…

Let $p>2$, and consider Lie algebras and Lie superalgebras with Cartan matrices whose entries lie in ${\mathbb Z}/p$. A Chevalley basis is the distinguished basis analogous to the…

Lie superalgebras with Cartan matrix have Chevalley generators and defining relations of Serre and non-Serre type. The non-Serre relations may be complicated to express in terms of…

Cocycle completeness conjecture. There are no more cocycles beyond those displayed in the table.

Generation conjecture. The components $\mathfrak{l}_0$, $\mathfrak{l}_2$, and $\mathfrak{l}_3$ generate $\mathfrak{l}_{\mathbin{\text{\raise.4ex\text{\bf.}}}}$.

Vanishing conjecture. For $\mathfrak{g}=\mathfrak{ag}_2$ and $\mathfrak{ab}_3$, and also for $\mathfrak{g}=\mathfrak{vect}(0|n)$ with $n>1$, one has

Associated-variety conjecture. For any integral dominant $\lambda\in\frak h^*$ with degree of atypicality $k$,

Two-dimensionality conjecture. No new simple symplectic triple systems appear in characteristic different from $3$, and every simple symplectic triple system in characteristic $3$…

Let ${\mathfrak {gl}}_{m|n}$ be the general linear Lie superalgebra, let $\kac{\lambda}$ be a Kac-module, and let permissible codes be the combinatorial objects defined in Definiti…

Let $\mathfrak{g}$ be a simple finite-dimensional Lie algebra, let $\mathfrak{n}$ be a nilpotent subalgebra, and write $C^j$ and $H^j$ for the corresponding cochain and cohomology…