Quasi-Hereditary Algebras and Quadratic Forms

✍ Scribed by Bangming Deng

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 165 KB
• Volume: 239
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.2000.8691

✦ Synopsis

The fundamental theorem on representation-finite quivers in 6 indicates a close connection between the representation type of a quiver and the definiteness of a certain quadratic form. Later a similar connection was discovered in other classification problems of representation theory. It turns out that there is a strong interaction of quadratic forms and the representation theory of finite-dimensional algebras.

Given a quasi-hereditary algebra A, instead of the complete module Ž . category, one studies the ⌬-good module category F F ⌬ consisting of A-modules which have a filtration by standard modules. Analogously to a Ž complete module category, one can associate quadratic forms also called . Ž . the Euler form and Tits form with the ⌬-good module category F F ⌬ . The aim of this paper is to study ⌬-good module categories in terms of these forms.

First, we study ⌬-directing and ⌬-omnipresent modules in the ⌬-good Ž . module category F F ⌬ over a quasi-hereditary algebra A and show that Ž . the existence of a ⌬-directing and ⌬-omnipresent module in F F ⌬ implies that all standard modules have projective dimension at most 2. By using w x the process of standardization introduced in 5 , the study of ⌬-directing Ž . modules in F F ⌬ can be reduced to the study of those over certain quasi-hereditary algebras which admit a ⌬-directing and ⌬-omnipresent module. For these quasi-hereditary algebras the quadratic forms associated with their ⌬-good module categories are well behaved. By applying Ž . Ž this reduction, we show that F F ⌬ is finite i.e., there are only finitely Ž .. many isomorphism classes of indecomposables in F F ⌬ if all indecompos-Ž .