Peter Jørgensen (Aarhus University): The green groupoid and its action on derived categories (joint work with Milen Yakimov)

Tuesday 13 October 2020

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We introduce the *green groupoid* G of a 2-Calabi–Yau triangulated category C. It is an augmentation of the mutation graph of C, which is defined by means of silting theory.

The green groupoid G has certain key properties:

- If C is the stable category of a Frobenius category E, then G acts on the derived categories of the endomorphism rings E(m,m) where m is a maximal rigid object.
- G can be obtained geometrically from the g-vector fan of C.
- If the g-vector fan of C is a hyperplane arrangement H, then G specialises to the Deligne groupoid of H, and G acts faithfully on the derived categories of the endomorphism rings E(m,m).

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