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arXiv · 2411.12912

Triangular decompositions: Reedy algebras and quasi-hereditary algebras

Teresa Conde·Georgios Dalezios·Steffen Koenig
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Abstract

Finite-dimensional Reedy algebras form a ring-theoretic analogue of Reedy categories and were recently proved to be quasi-hereditary. We identify Reedy algebras with quasi-hereditary algebras admitting a triangular (or Poincar\'e-Birkhoff-Witt type) decomposition into the tensor product of two oppositely directed subalgebras over a common semisimple subalgebra. This exhibits homological and representation-theoretic structure of the ingredients of the Reedy decomposition and it allows to give a characterisation of Reedy algebras in terms of idempotent ideals occurring in heredity chains, providing an analogue for Reedy algebras of a result of Dlab and Ringel on quasi-hereditary algebras.

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Teresa Conde, Georgios Dalezios, Steffen Koenig. 2024-11-19. Triangular decompositions: Reedy algebras and quasi-hereditary algebras. https://doi.org/10.1090/proc%2F17769

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