32 problems

Goswami's quantum rigidity conjecture. A genuine compact quantum group cannot act faithfully on a connected classical compact space.

Let $A$ be a unital $C^*$-algebra with a free coaction $\delta$ of a nontrivial compact quantum group $(H, \Delta)$. Equip the join $A \circledast^{\delta} H$ with the induced free…

Kac type characterization conjecture. If

Noncommutative Borsuk–Ulam conjecture. There does not exist a $\mathbb{G}$-equivariant $*$-homomorphism

Anti-deformation conjecture. Any Woronowicz algebra is a dimension-preserving $R^+$-deformation of a Woronowicz-Kac algebra.

Finiteness conjecture. There are only finitely many Woronowicz algebras having a given $(R^+,list)$ invariant.

Let $G_q$ be a $q$-deformed classical compact Lie group, and let its dual discrete quantum group have Poisson boundary $\mathcal{B}$. A quantum flag manifold is the quantum analogu…

Let $O_N$ denote the classical orthogonal group, and let $O_N^*$ and $O_N^+$ denote its half-liberated and free orthogonal quantum-group liberations, respectively. A liberation of…

The relevant standard tori of an easy compact quantum group $G$ are considered in the generalized standard torus setting. The Fourier tori are the standard tori associated with the…

First-column-space conjecture. The map

The notation $H_N^\times$ and $U_N^\times$ refers to the respective quantum group families and series discussed in the source. A contravariant duality reverses the relevant morphis…

Conjecture I. It is not possible to have a smooth faithful action of a genuine compact quantum group on $C(M)$.

Let $M$ be a compact connected smooth manifold, and let a compact quantum group act smoothly and faithfully on $C(M)$. A compact quantum group is genuine if it is not of the form…

Let $A$ be a unital C-algebra equipped with a free action of a non-trivial compact quantum group $(H,\Delta)$, and let $A\circledast^\delta H$ be the equivariant noncommutative joi…

Let $\alpha$ and $\beta$ be faithful, centrally ergodic actions of compact quantum groups respectively on $C(X)$ and $C(Y)$, where $X$ and $Y$ are finite sets. For an action…

Let $\mathbb{G}$ be the compact quantum group under consideration, let $\widehat{\mathbb{G}}$ be its set of irreducible representations, and let $\Sigma=\{F_n\}$ be a sequence of f…

Let $\mathbb{G}$ be a compact quantum group, let $L^1(\mathbb{G})$ be its associated convolution algebra, and let $\mathcal{Z}(L^1(\mathbb{G}))$ denote its center. For each irreduc…

Let $G\subset U_N^+$ be a compact quantum group, and let $\widehat{G}$ denote its discrete dual. For each $Q\in U_N$, let $\Gamma_Q$ be the associated discrete quantum group. Polyn…

Let $G\subset U_N^+$ be a compact quantum group. Write $C_{max}(G)$ and $C_{min}(G)$ for its maximal and minimal Hopf $C^*$-algebras; $G$ is coamenable when the canonical quotient…

Let $G\subset U_N^+$ be a connected compact quantum group. Its central algebra $C(G)_{central}$ is the norm closure of the linear span of irreducible representation characters. For…

Let $\mathbb{G}_{1}$ and $\mathbb{G}_{2}$ be compact quantum subgroups of $S_{N}^{+}$. For each $i=1,2$, let $\chi_i$ denote the character of the fundamental representation of…

Let $A$ be a unital C-algebra with a free action of a non-trivial compact quantum group $(H,\Delta)$. Let $\delta$ denote the coaction on $A$, and let $A*_{\delta}H$ be the equivar…

Let $\mathbb{G}$ be a compact quantum group. Classification conjecture for wreath-product deformations. - If $\mathbb{G}$ is an $R^{+}$-deformation of…

Conjecture I. It is not possible to have smooth faithful actions of genuine compact quantum groups on $C(M)$ when $M$ is a compact connected smooth manifold.

Let $G$ be a semisimple compact Lie group with weight lattice $P$ and root lattice $Q$, and let $\bigwedge^3(P/Q)\ne0$. Let $\Phi$ be a $3$-cocycle on $P/Q$ that does not lift to a…