22 problems

Let $A$ and $B$ be Hopf ${\mathbb C}^*$-algebras associated to quantum permutation groups, and let $P(A)$ and $P(B)$ denote their corresponding spin planar algebras, or skeletons.…

The even part of the Haagerup subfactor planar algebra $H\mathcal{P}$ is a subfactor planar algebra lying in a one-parameter family of subfactor planar algebras. Haagerup family co…

Let $\mathcal{B}$ be a bicommutant category admitting an absorbing object whose endomorphism algebra is a hyperfinite $\mathrm{II}_\infty$ or $\mathrm{III}_1$ factor. Assume that ……

Let $\mathscr{P}_{\bullet}$ be an irreducible planar algebra with index $\delta^2$, where $\delta>0$. For positive operators $x,y\in\mathscr{P}_{2,\pm}$ and parameters…

Let $\cP_\bullet$ be a unitary 2-shaded planar algebra. A finite-index homogeneous connected $\rm II_1$ multifactor inclusion is an inclusion $A\subset B$ with finite Jones index a…

Let $\b5$ be the parameter of the one-parameter family of three-dimensional canonical fusion bialgebras in case II, with $\b5\rfloor 3$. A fusion bialgebra is subfactorizable if it…

Let $p$ be as above and consider the planar algebra associated with the restricted quantum group $\bar{U}_{q}(\mathfrak{sl}_{2})$. For $n\geq 2p-1$, let $\alpha_i$ and $\beta_i$ be…

A strongly amenable graded hyperfinite subfactor of type II$_1$ is a subfactor whose grading, amenability, hyperfiniteness, and type are as specified; a subfactor planar para algeb…

The $\mathscr{C}^{N,1,1}$ conjecture. The unitary fusion category $\mathscr{C}^{N,1,1}$ is isomorphic to Xu's bimodule category for this conformal inclusion. The source verifies th…

Let $N$ be a positive integer, let $\mathscr{C}^{N}_{\bullet}$ be the unshaded subfactor planar algebra, and let $g$ denote the grading object used to form its quotient modulo $g$.…

Let $G$ be a non-trivial finite group, let $\cR_\bullet$ be the reduced subfactor planar algebra of a $3^G$ subfactor planar algebra, and let $\cQ_\bullet$ be the planar subalgebra…

Enumeration conjecture. At index $3+\sqrt{5}$, there are exactly $13$ non-Temperley–Lieb subfactor planar algebras: the Bisch–Jones Fuss–Catalan $A_3*A_4$ subfactor planar algebra…

Morrison–Peters conjecture. There are exactly $2$ non-Temperley–Lieb subfactor planar algebras with index in $(5,3+\sqrt{5})$: the unique $\mathfrak{su}(2)_5$ subfactor planar alge…

Let ${\cal A}\cT_{n,\omega_U}$ denote the fusion category associated with the construction considered in the paper, indexed by an integer $n$ and a parameter $\omega_U$. The preced…

Let a subfactor planar algebra mean the planar algebra associated with a finite-index subfactor, and let $A_\infty$ denote the infinite-depth principal graph. Consider irreducible…

Morrison–Peters conjecture. Any subfactor with index in the range $(5,3+\sqrt{5})$ has principal graphs $(A_\infty,A_\infty)$, the pair of graphs represented in the source by…

Let $N\subset M$ be a finite-depth subfactor, let $P$ be its associated subfactor planar algebra, and let the $N$-$N$-bimodule category generated by $_N L^2(M)_M$ be the category w…

Let $\Lambda$ be a $\mathbb{C}$-linear category. Let $\Lambda^{\mathcal R}$ and $\Lambda^{\mathcal A}$ be, respectively, the rectangular and annular versions of the corresponding l…

Let $P^{\mathcal{E}^{(8)}}$ denote the object associated with the graph $\mathcal{E}^{(8)}$, and let the modules $H^{\beta_{(0,0)}}$, $H^{\beta_{(3,0)}}$, $H^{\beta_{(0,3)}}$,…

Let $G_n$ be an easy quantum group and $G_n^+$ its liberation, with inclusion $G_n\subset G_n^+$. This inclusion gives a planar algebra module in the sense of Jones, and the Gram m…

Let $\mathcal{P}$ be a fixed finite-depth planar algebra, let $V_{\beta}=\cup+\beta W\in\mathcal{P}$ be a family of elements, and let $\tau_{\beta}=\tau_{V_{\beta}}$. Denote by…

Let $1$ denote the trivial representation of $SU(3)$, let $\rho$ be the fundamental representation, and let $\overline{\rho}$ be its conjugate representation. For nonnegative integ…