Derived Picard Groups of Finite-Dimensional Hereditary Algebras

Jun Ichi Miyachi, Amnon Yekutieli

Research output: Contribution to journalArticlepeer-review

44 Scopus citations

Abstract

Let A be a finite-dimensional algebra over a field k. The derived Picard group DPick(A) is the group of triangle auto-equivalences of Db(modA) induced by two-sided tilting complexes. We study the group DPick(A) when A is hereditary and k is algebraically closed. We obtain general results on the structure of DPick(A), as well as explicit calculations for many cases, including all finite and tame representation types. Our method is to construct a representation of DPick(A) on a certain infinite quiver Γirr. This representation is faithful when the quiver A of A is a tree, and then DPick(A) is discrete. Otherwise a connected linear algebraic group can occur as a factor of DPick(A). When A is hereditary, DPick(A) coincides with the full group of k-linear triangle auto-equivalences of Db(modA). Hence, we can calculate the group of such auto-equivalences for any triangulated category D equivalent to Db(modA). These include the derived categories of piecewise hereditary algebras, and of certain noncommutative spaces introduced by Kontsevich and Rosenberg.

Original languageEnglish
Pages (from-to)341-368
Number of pages28
JournalCompositio Mathematica
Volume129
Issue number3
DOIs
StatePublished - 1 Dec 2001

Keywords

  • Derived category
  • Finite dimensional algebra
  • Picard group
  • Quiver

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