8 problems

Cuntz's doubled-rank free-algebra conjecture. The multiplication on $A'$ given by $xy=\varphi(\pi(x)\pi(y))$ is associative and has structure constants in $\mathbb N$. Equivalently…

Let $d\geq 1$, let $1\leq n\leq d$, and let $I_{n,d}$ be the set of strictly increasing $n$-tuples $(i_1,\ldots,i_n)$ with $0\leq i_1<\cdots<i_n<d$. Set $\xi=\exp(i\pi/d)$,…

Let $\mathfrak{g}$ be a nilpotent FC set, and let $d$ be an integer dividing $\cs{\du{\mathfrak{g}}}$. An FC set $\mathfrak{h}\subseteq\mathfrak{g}$ is a sub-FC set of…

Let $\mathcal{H}^{ab}_{i}$ and $\mathcal{H}^{bc}_{j}$ be Hilbert spaces associated with chiral sectors and boundary conditions, and let their fusion product be defined by the quoti…

Cuntz's conjecture. This multiplication is associative.

Special case of the Level Increasing Conjecture. Under these hypotheses,

Conjectural radical presentation. For $\mathfrak{g}$ of type $B_r$ with $r\geq 3$,

Let $t$ be an involutive automorphism of a self-dual vertex operator superalgebra $W$. The fixed-point orbifold conjecture. The fixed-point subalgebra $W^{\langle t\rangle}$ is rat…