On representation-finite algebras and beyond
Klaus Bongartz
TL;DR
This paper surveys the theory of representation-finite and minimal representation-infinite algebras, focusing on multiplicative bases, coverings, and applications like the Brauer-Thrall conjecture and module length gaps.
Contribution
It introduces the use of ray-categories to achieve key properties and provides new proofs for longstanding conjectures in algebra representation theory.
Findings
Proof of a sharper second Brauer-Thrall conjecture
No gaps in lengths of indecomposable modules
Existence of multiplicative bases and coverings
Abstract
We give a survey on the theory of representation-finite and certain minimal representation-infinite algebras.The main goals are the existence of multiplicative bases and of coverings with good properties. Both are attained via ray-categories. As applications we include a proof of a sharper version of the second Brauer-Thrall conjecture and of the fact that there are no gaps in the lengths of the indecomposable modules over an algebra.
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Homotopy and Cohomology in Algebraic Topology · Advanced Algebra and Geometry