c-vectors of 2-Calabi-Yau categories and Borel subalgebras of sl infinity

Peter Jørgensen, Milen Yakimov

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Abstract

We develop a general framework for c-vectors of 2-Calabi–Yau categories, which deals with cluster tilting subcategories that are not reachable from each other and contain infinitely many indecomposable objects. It does not rely on iterative sequences of mutations. We prove a categorical (infinite-rank) generalization of the Nakanishi–Zelevinsky duality for c-vectors and establish two formulae for the effective computation of c-vectors—one in terms of indices and the other in terms of dimension vectors for cluster tilted algebras. In this framework, we construct a correspondence between the c-vectors of the cluster categories C(A ) of type A due to Igusa–Todorov and the roots of the Borel subalgebras of sl . Contrary to the finite dimensional case, the Borel subalgebras of sl are not conjugate to each other. On the categorical side, the cluster tilting subcategories of C(A ) exhibit different homological properties. The correspondence builds a bridge between the two classes of objects.

Original languageEnglish
Article number1
JournalSelecta Mathematica
Volume26
Issue1
ISSN1022-1824
DOIs
Publication statusPublished - Feb 2020
Externally publishedYes

Keywords

  • 2-Calabi–Yau category
  • Cluster category of type A
  • Homological index
  • Kac–Moody algebra
  • Levi factor
  • g-vector

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