17 problems

Let $ell$ be an admissible level for the simple Lie algebra $\bf g$. Vertex tensor and ribbon category conjecture. 1. $L_{ell}(\bf g)\operatorname{-wtmod}$ is a vertex tensor categ…

Grothendieck's topological Teichmüller tower conjecture. The duplication functor $dupl:RG\to {\cal S}$ is an equivalence of categories.

Let $F4(d)$ be the symmetric ribbon category generated by a trivalent vertex subject to the defining $F4(d)$ relations, and let $\operatorname{Hom}$ denote a morphism space in this…

Let $U$ be a subset of $\mathbb C$, and let the Cauchy completion mean the additive and idempotent completion of a category. Classical semisimplicity conjecture. There is a dense s…

Let $R$ be a ring and let $\boldsymbol{\lambda}$ be a parameter such that the category $\mathsf{Exc}_{R,\lambda}$ is defined; write $\mathsf{Exc}_{R,\lambda}(n)$ for its space of d…

A symmetric ribbon category is a monoidal category equipped with symmetry and ribbon structure, and a one-parameter family of Lie algebra objects is a family parametrized by a cont…

The BQ(r) module-category conjecture. $\operatorname{Rep}^0\mathfrak{B}_{rP}^{a_r}$ is non-degenerate and ribbon if $r$ is odd or $\rho\in Q$, and its irreducible modules are

The conjectures. $\operatorname{Rep}_{\langle s\rangle}\mathcal{W}_Q(r)$ is a finite non-degenerate ribbon category with…

Let $\mathcal{M}(p)$ be the singlet vertex operator algebra, and let $\mathrm{\bf Rep}\,\mathcal{M}(p)$ be its representation category equipped with the $P(1)$-tensor product. The…

Let $\mathcal{V}_{\mathrm{ev}}(N)$ be the even part of the super-vertex operator algebra of $N$ pairs of symplectic fermions, with central charge $c=-2N$. Let $\mathsf{Q}(N,\beta)$…

Let $\overline{U}_{q}^{H}s\ell(2)$ be the unrolled restricted quantum group, and let $\mathrm{\bf Rep}_{\!\mathrm{wt}\,}\overline{U}_{q}^{H}s\ell(2)$ denote its finite-dimensional…

Let $p\geq 2$, let $q=e^{\pi i/p}$, let $\overline{U}^{(\Phi)}_{q} s\ell(2)$ be the factorisable ribbon quasi-Hopf algebra described above, and let $\mathcal{W}(p)$ be the triplet…

Let $\mathrm N$ be the parameter specifying the number of symplectic fermion pairs, let $\beta$ be the scalar used in the ribbon structure, and let…

Ribbon category conjecture. For even ordered $\xi$, there exists a category $\mathscr{C}^H_{even}$ of modules over ${{\mathcal U}^{H}}_\xi({\mathfrak{g}})$ which is ribbon.

Let $\mathcal{S}\mathcal{F}(\mathfrak{h})$ be the finite ribbon category associated with symplectic fermions and let $\operatorname{Rep}(\mathcal{V}_{\mathrm{ev}})$ be the represen…

The quantised Vogel plane is a finitely presented ribbon category defined over the field of rational functions in $q^\alpha,q^\beta,q^\gamma$, where these indeterminates and the as…

Let $\mathcal{D}^{H}$ be the category of modules under consideration, and let $\varpi\in\mathcal{U}^{H}$ be the central element defined by $\varpi=uS(u)^{-1}K_{2\rho}^{2(r-1)}$. Le…