12 problems

Rouquier's conjecture. A subcategory affording an abelian $2$-representation should exist.

Let $\sc\text{B}\hspace{1.0pt}_{n}^{\operatorname{str}}$ be the finitary bicategory under consideration, and let simple transitive $2$-representations of it be considered up to equ…

Let $n\geq 1$, and let $\mathcal{C}_n$ be the left cell $2$-subcategory of projective functors for the star algebra considered in the paper. A simple transitive $2$-representation…

A simple transitive $2$-representation is a simple transitive action of the relevant $2$-subcategory of projective functors on a finitary category, and a cell $2$-representation is…

2-McKay conjecture. For each character $\chi$, the 2-representations $\Theta_H(\chi)$ and $\Theta_N(\chi)$ of $G$ are isomorphic. This is a 2-representation-theoretic strengthening…

Let $\mathfrak{g}$ be a simple finite-dimensional complex Lie algebra, let $\chi\in\mathfrak{g}^*$ be its nilpotent character, and let $G$ be the associated finite component group.…

Let $\mathcal{K}$ be a crossed module with finite fundamental group $\pi_1(\mathcal{K})$, let $\widetilde{\mathcal{K}}$ denote its associated 2-group, and let $\mathcal{X}$ be a cl…

Let $\mathcal{K}=(H\xrightarrow{\partial}G)$ be a crossed module whose fundamental group $\pi_1(\mathcal{K})$ is finite. Let $\Theta$ be a 2-representation of…

Semisimplicity conjecture. If each two-sided cell of $\mathcal{C}$ is idempotent, then the diagram of any isotypic $2$-representation of $\mathcal{C}$ is semi-simple.

Let $\widehat{\widehat{\mathfrak{g}}}$ be the double-affine algebra in the sense of the paper, and let $B_{\widehat{\widehat{\mathfrak{g}}}}$ be its double affine braid group. A do…

Let $\mathfrak{g}$ be a Kac–Moody Lie algebra whose Dynkin diagram is simply laced and contains no loops. A geometric categorical $\mathfrak{g}$ action is the categorical action on…

The 2-category $\mathcal{U}_Q(\mathfrak{g})$ is the categorified quantum group associated with a Kac–Moody Lie algebra $\mathfrak{g}$. A 2-representation is an action of this 2-cat…