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SUMMARY:Peter Jørgensen (Aarhus University)
DTSTART:20201013T150000Z
DTEND:20201013T160000Z
DTSTAMP:20260423T035417Z
UID:OCAS/7
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/OCAS/7/">The
  green groupoid and its action on derived categories (joint work with Mile
 n Yakimov)</a>\nby Peter Jørgensen (Aarhus University) as part of Online 
 Cluster Algebra Seminar (OCAS)\n\n\nAbstract\nWe introduce the green group
 oid $\\mathcal{G}$ 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.\n\nThe green groupoid $\\mathcal{G}
 $ has certain key properties:\n\n1. If $\\mathcal{C}$ is the stable catego
 ry 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.\n\n2. $\\mathcal{G}$ can be obtained geometr
 ically from the $g$-vector fan of $\\mathcal{C}$.\n\n3. 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 endomorphis
 m rings $\\mathcal{E}(m\,m)$.\n\nThe 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 singul
 arity.\n
LOCATION:https://researchseminars.org/talk/OCAS/7/
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