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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 2 / Oct 10, 2026 / CC BY 4.0

Abstract

Let AA be a representation-infinite finite-dimensional algebra over an algebraically closed field kk. We prove that for every infinite cardinal κ\kappa there is an indecomposable AA-module MM with dim⁡kM=κ\dim_k M=\kappa. Over algebraically closed fields, this settles Simson's conjecture, also called the third Brauer--Thrall conjecture, that every representation-infinite finite-dimensional algebra has indecomposable modules of arbitrarily large cardinality. To construct these modules, we extend Bongartz's representation embedding from finite-dimensional Kronecker modules without a simple injective summand to such modules of arbitrary dimension. The extended functor 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 yields the required AA-modules.

Provenance statement

Proof and almost all math arguments: Astra Paper revisions: Astra main writing, Opus reviewers, Argon wording polish, using https://github.com/haruhisa-enomoto/math-paper-skills Formalization: Opus mainly, for some points Astra helped Opus

Tools used

OpenAI
CodexVersion GPT-6-Astra
Anthropic
Claude OpusVersion Opus 5.5
Google
GeminiVersion Argon 4

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Version history

  1. v1Submitted by Haruhisa EnomotoInitial depositSupersededOct 10, 2026
  2. v2Submitted by Haruhisa EnomotoSubmitted as a new versionCurrentNov 04, 2026