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Representation embeddings and the third Brauer--Thrall conjecture

Primarily AI-generated textHuman understanding: some partsmath.RT — Representation Theorymath.RA — Rings and Algebras

Contributed by Haruhisa Enomoto ↗

Version 1 / Oct 10, 2026 / CC BY 4.0

Abstract

We show that every representation-infinite finite-dimensional algebra over an algebraically closed field has an indecomposable module of each infinite dimension. This settles, over algebraically closed fields, Simson's conjecture, also called the third Brauer--Thrall conjecture, that such an algebra has indecomposable modules of arbitrarily large cardinality. To construct these modules, we extend a representation embedding of Bongartz from finite-dimensional Kronecker modules without a simple injective summand to a representation embedding of such Kronecker modules of arbitrary dimension, which commutes with direct sums and filtered colimits and preserves the dimension of infinite-dimensional modules. Applying it to the indecomposable Kronecker modules of every infinite dimension constructed by G\"obel and Simson gives the required modules.

Provenance statement

The research was done under the author's direction using GPT-6-Astra, which found and developed the proof presented here and wrote the earlier drafts of the manuscript. The present text was revised with Claude Opus~5.5. The author directed the research and revisions and is responsible for the mathematical content and the final manuscript.

Tools used

OpenAI
CodexVersion GPT-6-Astra
Anthropic
Claude OpusVersion Opus 5.5

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Version history

  1. v1Submitted by Haruhisa EnomotoInitial depositCurrentOct 10, 2026