31 problems

Let $(V,c)$ be a braided vector space of diagonal type over an algebraically closed field of characteristic zero, and let $\mathfrak{B}(V)$ denote its Nichols algebra. The finite G…

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 $(V,c)$ 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…

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…

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

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 $\mathcal{O}$ 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…

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 $V$ 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…

Orbit-profile invariance conjecture. Equality of the orbit profiles should preserve the stated dimension, or Gelfand–Kirillov dimension, under the conditions intended in the source…

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

Angiono–Azevedo–Heckenberger conjecture. If $\mathscr{B}_{\mathfrak{q}}$ has finite Gelfand–Kirillov dimension, then its root system is finite.

Integral conjecture. The element

Let $\mathbb{S}_m$ be the symmetric group, and let $\EuScript{B}_{\mathbb{S}_m}$ be its Nichols–Woronowicz algebra model for Schubert calculus. Infinite-dimensionality conjecture.…

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…

Let $\boldsymbol{G}$ be a finite simple non-abelian group. A complex Yetter–Drinfeld module $V$ of $\boldsymbol{G}$ has an associated Nichols algebra $\mathfrak B(V)$. Finite simpl…

Let $L$ be a cosemisimple Hopf algebra and let $A$ be a pointed Hopf algebra with diagram $\mathcal R$ and infinitesimal braiding $\mathcal V=\mathcal R^1$ in…

Let $\mathfrak{g}$ be a Lie algebra, let $ell=4$, and let $\theta$ be a diagram automorphism. Consider an orbifold model of the logarithmic conformal field theory for…

Let $\Lambda$ be a fractional lattice with short basis $\alpha_i^{short}$, and let $\Lambda^{long}$ be the integral sublattice with associated long basis $\alpha_i^{long}$, chosen so t…

Let $M\langle(xy,x)\rangle$, $M\langle(y,xy)\rangle$, and $W^{b_1,-1}$ denote the Yetter–Drinfeld modules introduced in the paper, and let $\mathfrak{B}(-)$ denote their Nichols al…

Let $G=G(m,p,n)$ be a complex reflection group in the non-exceptional series, and let $\mathcal{B}_G$ be its elementary Fomin–Kirillov Nichols algebra. Finite-dimensionality conjec…