Bilinear forms on Grothendieck groups of triangulated categories

TL;DR

This paper generalizes bilinear forms on Green rings to the Grothendieck groups of triangulated categories with Auslander-Reiten triangles, exploring their properties and applications to homotopy categories of perfect complexes.

Contribution

It extends the theory of bilinear forms to triangulated categories, introduces a Hermitian form, and applies it to categories of perfect complexes over symmetric algebras.

Findings

The bilinear form can be modified to a Hermitian form with a dual basis.

The form's non-degeneracy depends on the category's properties.

Application to the homotopy category of perfect complexes over symmetric algebras.

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.