66 problems

Let $K=\Bbbk G$ for a finite group $G$, and let … be a finite-dimensional coradically graded connected Hopf algebra, with $R^1$ its degree-one component. Andruskiewitsch–Schneider…

Let $W$ be a vertex operator algebra containing Heisenberg fields $\alpha_i$, and let $S_i=\operatorname{Res}(e^{\alpha_i})$. Assume that $W$ decomposes into integer eigenvalues un…

Let $\EuScript{E}_m$ be the Fomin–Kirillov algebra over a field $\mathbf{k}$ associated with the symmetric group $\mathbb{S}_m$. Fomin–Kirillov's conjecture. The algebra…

Heckenberger–Lochmann–Vendramin conjecture. Any finite-dimensional elementary Nichols algebra of group type is $bg$-equivalent to one of those listed in Table 9.1 of Heckenberger's…

Let $\mathcal{O}$ be a finite rack of type C, let $\mathfrak{q}$ be a faithful $2$-cocycle on $\mathcal{O}$, and let $\mathscr{B}(\mathcal{O},\mathfrak{q})$ denote the associated N…

Let $\Gamma$ be an abelian finitely generated group, let $V$ be a Yetter–Drinfeld module of diagonal type over $\Gamma$, and let…

Let $n\geq 3$. Let $\mathcal{FK}_n$ be the quadratic algebra generated by the vector space $V_n$ with basis $\{x_{(ij)}\mid 1\leq i<j\leq n\}$, equipped with its graded bialgebra s…

A pointed Hopf algebra is a Hopf algebra whose coradical is a group algebra. The class under consideration consists of Hopf algebras generated by skew-primitive elements and an abe…

The isomorphism conjecture. The algebra homomorphism $\eta$ is an isomorphism. The statement appears as a conjecture in the source, but the supplied material gives no resolution ev…

Let $Fl_n$ be the flag variety of type $A$, let $\widetilde{{\mathcal{E}}}_n$ be the algebra over $\mathbb{Z}[q_1,\ldots,q_{n-1}]$ obtained from ${\mathcal{E}}_n$ by replacing rela…

Let $(V,c)$ and the matrix $(A_{ij})$ be as above: $V=\mathbb C x_1\oplus\cdots\oplus\mathbb C x_{\theta}$, $c(x_i\otimes x_j)=q_{ij}x_j\otimes x_i$, $q_{ii}=-1$,…

Let $(V,c)$ be a braided vector space with $V=\mathbb C x_1\oplus\cdots\oplus\mathbb C x_{\theta}$ and … Assume that $q_{ii}=-1$ for every $i$ and $q_{ij}\in\{\pm1\}$ for all $i,j$…

Let $G$ be a finite group. Say that $G$ collapses if every finite-dimensional pointed Hopf algebra $H$ with group of grouplikes $G(H)\simeq G$ is isomorphic to $\Bbbk G$; equivalen…

Let $\mathcal{V}$ be a vertex algebra, let $\mathfrak{Z}_{\mathcal{M}_1},\ldots,\mathfrak{Z}_{\mathcal{M}_n}$ be screening operators, and let $\mathcal{W}\subset\mathcal{V}$ be the…

A vertex algebra $\mathcal{V}$ has nonlocal screening operators arising from modules $\mathcal{M}\neq \mathcal{V}$; these operators may involve non-integral powers of the integrati…

Finite generation conjecture. The finite-dimensionality of $\mathfrak B$ should imply that $\operatorname{Ext}_{\mathfrak B}(k,k)$ is finitely generated.

Logarithmic Kazhdan–Lusztig conjecture. In good cases, there is an equivalence of braided tensor categories

Let $G$ be a finite simple non-abelian group, and let $D(G)$ be its Drinfeld double. A finite-dimensional Nichols algebra associated with a $D(G)$-module is a Nichols algebra arisi…

Let $M(p)$ be the singlet VOA, let $H_\phi$ be its Heisenberg free-field realization, and let $A$ be the algebra object defined by $H_\phi$ in the category of $M(p)$-modules. Let…

Let $K$ be a finite-dimensional cosemisimple Hopf algebra, let $V\in {}^{K}_{K}\mathcal{YD}$ be a Yetter–Drinfeld module such that $\dim \mathscr{B}(V)<\infty$, and let…

Nichols-algebra and screening-operator conjecture. The screening operators generate a corresponding Nichols algebra $\mathfrak{N}$ in the braided tensor category $\mathcal{C}$;…

Let $d\in\mathbb{N}$ and let ${\bf q}\in\mathbb{C}^{\{1,2\}^d}$ be a $d$-dimensional tensor defining a Weyl groupoid through the generalized Rosso condition. Rank-two realization c…

Let $d$ be an even tensor power and let the generalized Rosso condition associate a Weyl groupoid to a higher diagonal braiding. Asymptotic realization conjecture. Every finite Wey…

Let $N_t$ be an object in the class whose existence is asserted, let $V$ be the associated Yetter–Drinfeld module, and let $V_0$ denote its degree-zero component. For $k\ge 2$, let…

Let $V$ be a braided vector space of diagonal type, and let $\mathscr{B}(V)$ be its Nichols algebra. The root system of $\mathscr{B}(V)$ is the root system arising from its restric…