Bilinear Forms on Grothendieck Groups of Triangulated Categories

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

We extend the theory of bilinear forms on the Green ring of a finite group developed by Benson and Parker to the context of the Grothendieck group of a triangulated category with Auslander–Reiten triangles, taking only relations given by direct sum decompositions. We examine the non-degeneracy of the bilinear form given by dimensions of homomorphisms, and show that the form may be modified to give a Hermitian form for which the standard basis given by indecomposable objects has a dual basis given by Auslander–Reiten triangles. An application is given to the homotopy category of perfect complexes over a symmetric algebra, with a consequence analogous to a result of Erdmann and Kerner.

Original languageEnglish (US)
Title of host publicationGeometric and Topological Aspects of the Representation Theory of Finite Groups - PIMS Summer School and Workshop, 2016
EditorsSrikanth B. Iyengar, Julia Pevtsova, Jon F. Carlson
PublisherSpringer New York LLC
Pages465-480
Number of pages16
ISBN (Print)9783319940328
DOIs
StatePublished - 2018
EventPIMS Summer School and Workshop on Geometric Methods in the Representation Theory of Finite Groups, 2016 - Vancouver, Canada
Duration: Jul 27 2016Aug 5 2016

Publication series

NameSpringer Proceedings in Mathematics and Statistics
Volume242
ISSN (Print)2194-1009
ISSN (Electronic)2194-1017

Other

OtherPIMS Summer School and Workshop on Geometric Methods in the Representation Theory of Finite Groups, 2016
CountryCanada
CityVancouver
Period7/27/168/5/16

Bibliographical note

Funding Information:
The author was supported by Simons Foundation award 282425.

Keywords

  • Auslander–Reiten triangle
  • Green ring
  • Perfect complex
  • Symmetric algebra

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