31 problems

Triviality conjecture. Any semisimple Hopf algebra $H$ of dimension $pq$ over $k$ is trivial.

Let $H$ be a Hopf algebra over the underlying field, and suppose that its dimension is a prime number. Kaplansky's eighth conjecture. Then $H$ should be commutative and cocommutati…

The structure matrices $A,B,C,D$ satisfy the displayed cubic consistency relations and generalized unitarity conditions, and the dual exchange relation defines a trace construction…

Dimension-$p^3$ classification conjecture. Every Hopf algebra $H$ of dimension $p^3$ is semisimple, pointed, or has pointed dual; equivalently, $H$ is one of the Hopf algebras in t…

Let $U_0$ be the subspace used in the paper's decomposition associated with a tridiagonal pair, and let $L$ and $R$ denote the corresponding lowering and raising maps. Commutativit…

Masuoka's conjecture. Up to quasi-isomorphism, there should be only finitely many Hopf algebras of any given dimension.

Finiteness conjecture. In any given dimension, the number of non-isomorphic weak Hopf algebras $H$ with this property is finite.

Let $R$ be a unitary involutive $R$-matrix. An invertible matrix $\alpha$ is assumed to satisfy the relation referred to as Eq. (Ralphadef-graphical), and $\theta$ denotes the part…

Separator conjecture. The object $0$ is a separator of $\mathsf{XC}$.

Let $\mathcal{C}$ be a modular, ribbon or braided fusion category over $\mathbb{C}$. A category is Hermitian over a field $K$ when it admits the corresponding Hermitian structure d…

Let $N$ and $n$ be parameters for the skein algebra $\mathrm{SkAlg}_{\mathrm{SL}_N,n}$, and let $\overline{\pi}$ denote the element defined in the surrounding discussion. Central p…

A KV solution is a solution of the Kashiwara–Vergne equations, and an associator is a Drinfel'd associator satisfying the pentagon and hexagon equations. Alekseev–Torossian conject…

Large-$N$ commutation-relation conjecture. In the large $N$ limit, these operators satisfy

Let $p\geq 2$, set $\epsilon_1=1$, $\epsilon_2=k+p$, and define the rescaled operators … Let $\mathcal B$ be an appropriate energy eigenbasis for the Hilbert space. Non-Abelian lar…

Let $\mathscr O^{(p)}_\infty$ be the large $N$ limit algebra of quantum observables, and let $\mathcal H_\infty$ be the large $N$ limit Hilbert space of the Chern–Simons matrix mod…

Associativity conjecture. The structure $(A,*,^{-1})$ is an associative invertible magmoid. After adjoining the necessary units, $(A,*,^{-1})$ forms a groupoid.

Pivotality conjecture. Every semisimple weak Hopf algebra is pivotal.

Let $X$ be the underlying finite set, let $F$ be the twist element, and let $\Delta$ be the coproduct on the associated algebra. Write $F^*_{12,3}$ for the element defined earlier…

What is lacking in the previous list is the generalization $\mathbf{saw}(n)$ of the algebra $\mathbf{saw}(3)$. Such a generalization would provide a description of the algebra…

Modular-data classification conjecture. The modular data, together with the topological central charge, provide a gauge-invariant means to classify modular tensor categories.

Let $X$ be an object of a braided fusion category, and let $B_n$ act on $X^{\otimes n}$ through the braiding, for each integer $n\geq 1$. Write $\operatorname{FPdim}(X)$ for the Fr…

Specialization conjecture. Under the algebraic specialization $\wp(x)=x^n$, the groups

Equivalence conjecture. The modular functor induced by $\mathrm{V}'_{\mathcal C}$ is equivalent to Lyubashenko's modular functor.

Let $\mathscr A$ be the algebra of quantum hypermatrices, with quantum hyperdeterminant $\mathrm{Det}_q$. An intermediate quotient algebra is a quotient…

Weak-integrality conjecture. These braid group representations have finite image for all $n>0$ if and only if $\operatorname{FPdim}(X)^2\in\mathbb Z$.