The green groupoid and its action on derived categories (joint work with Milen Yakimov)

Abstract: We introduce the green groupoid $\mathcal{G}$ of a $2$-Calabi-Yau triangulated category $\mathcal{C}$. It is an augmentation of the mutation graph of $\mathcal{C}$, which is defined by means of silting theory.

The green groupoid $\mathcal{G}$ has certain key properties:

1. If $\mathcal{C}$ is the stable category of a Frobenius category $\mathcal{E}$, then $\mathcal{G}$ acs on the derived categories of the endomorphism rings $\mathcal{E}(m,m)$ where $m$ is a maximal rigid object.

2. $\mathcal{G}$ can be obtained geometrically from the $g$-vector fan of $\mathcal{C}$.

3. If the $g$-vector fan of $\mathcal{C}$ is a hyperplane arrangement $\mathcal{H}$, then $\mathcal{G}$ specialises to the Deligne groupoid of $\mathcal{H}$, and $\mathcal{G}$ acts faithfully on the derived categories of the endomorphism rings $\mathcal{E}(m,m)$.

The situation in (3) occurs if $\Sigma_{\mathcal{C}}^2$, the square of the suspension functor, is the identity. It recovers results by Donovan, Hirano, and Wemyss where $\mathcal{E}$ is the category of maximal Cohen-Macaulay modules over a suitable singularity.

mathematical physicscommutative algebraalgebraic geometrycombinatoricsquantum algebrarings and algebrasrepresentation theory

Audience: researchers in the topic

Online Cluster Algebra Seminar (OCAS)