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Representation embeddings and the third Brauer--Thrall conjecture

Haruhisa Enomoto

Hexagonhexagon/2610.00179Oct 10, 2026math.RTAI-generatedscore 0

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,…

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