67 problems

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…

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…

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

The atypicality conjecture. For every basic classical Lie superalgebra ${\mathfrak{g}}$ and every simple supermodule $L(\lambda)$ in $\mathcal{F}$,

Let ${\mathfrak{g}}$ be a simple basic classical Lie superalgebra, and let $L(\lambda)$ be a finite-dimensional simple ${\mathfrak{g}}$-supermodule. Write…

A symplectic triple system over a field of characteristic $3$ is called null when its alternating bilinear form is zero. Dimension conjecture. The dimension of any simple null symp…

Let ${\mathfrak{g}}$ be a basic classical Lie superalgebra, and let $L(\lambda)$ be a simple ${\mathfrak{g}}$-supermodule of highest weight $\lambda$. Write…

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…

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

Character map conjecture. If $i:K(\mathcal O_{m|n}^\Delta)\to\mathscr T^{m|n}_{\mathbb Z}$ is the map defined in the cited theorem, then

Let $p,q$ be positive integers, set … and for integers $m,n$ set … For the Lie superalgebra ${\mathcal L}_s$ and the irreducible module $L(c_{p,q},h_{p,q}^{m,n})$, finite-dimension…

Let $W(n)$ be the Lie superalgebra of type $W(n)$, and let its blocks be the blocks of the category of finite-dimensional representations. A block is nontrivial if it does not cont…

Let $\mathrm{osp}(1|2)$ act on the space ${\cal D}_{\lambda,\mu}$ of differential operators from tensor densities of weight $\lambda$ to tensor densities of weight $\mu$, and let…

Let $\mathfrak{g}$ be an (almost) simple finite-dimensional Lie superalgebra of positive defect, namely one of $\mathfrak{sl}(m,n)$ with $m,n\geq 1$, $\mathfrak{osp}(m,2n)$ with…

Let $B(m,n)=\mathfrak{osp}(2m+1|2n)$ with $m+n\geq 2$, and let $\mathrm{gb}$ denote the collection of deformation parameters in the $\gamma$-deformation. General $B(m,n)$ conjectur…

Let ${\mathfrak g}$ be a Lie superalgebra, ${\mathfrak l}$ a subalgebra, $M$ a ${\mathfrak g}$-supermodule, and $x\in {\mathcal Y}_{{\mathfrak l}}$. Write…

Let $cmathfrak{gl}(1|1)$ be the smallest type-A Lie superalgebra, let $cmathcal V_{chbar,k}(cmathfrak{gl}(1|1))$ be its quantum vertex algebra, and set … Here $csigma^{n,0}$ denote…

Let $\mathfrak{sl}(m|n)$ be the type $A$ Lie superalgebra, let $V_{(2s)}$ be its covariant representation, and write … where $\lambda$ ranges over $(m,n)$-hook Young diagrams. Deco…

Let $L=L_\mathbf{0}\oplus L_\mathbf{1}$ be a quasireductive Lie superalgebra, and let an almost inner derivation of $L$ be a derivation that is almost inner, while an inner derivat…

Quantum toroidal superalgebra isomorphism. There is an isomorphism

Let $D(2,1;\lambda)$ be the exceptional Lie superalgebra with Dynkin diagrams $G(D_{\lambda})$ and $G(D_{\lambda}^{\circ})$ related by the odd reflection $s_{\alpha_2}$, and let…

Mixed tensor product conjecture. There is an isomorphism of $U(\mathfrak{gl}(m+1|n))$-modules

Vanishing conjecture. For all $m\geq 1$, if $K$ is a slice virtual knot, then

Let $\mathfrak{g}$ be a simple, finite-dimensional, basic contragredient Lie superalgebra whose even subalgebra is reductive. Let $U_hmathfrak{g}$ be its quantized universal envelo…