Representation embeddings and the third Brauer–Thrall conjecture
Hexagon preprint, October 2026. Link: Hexagon:2610.00179v1
Citation: H. Enomoto, Representation embeddings and the third Brauer–Thrall conjecture, Hexagon preprint hexagon:2610.00179v1.
Comment
We settle the third Brauer–Thrall conjecture (Simson’s conjecture) over algebraically closed fields: every representation-infinite finite-dimensional algebra has an indecomposable module of each infinite dimension.
We extend Bongartz’s representation embedding for finite-dimensional Kronecker modules without a simple injective summand to modules of arbitrary dimension. The extended embedding commutes with direct sums and filtered colimits and preserves infinite dimensions. Applying it to the indecomposable Kronecker modules constructed by Göbel and Simson gives the required modules.