Triangular decompositions: Reedy algebras and quasi-hereditary algebras

TL;DR

This paper characterizes finite-dimensional Reedy algebras as quasi-hereditary algebras with a specific triangular decomposition, linking their structure to tensor products of subalgebras and idempotent ideals.

Contribution

It provides a new characterization of Reedy algebras via triangular decompositions and hereditary chains, connecting them to known quasi-hereditary algebra structures.

Findings

Reedy algebras are identified with quasi-hereditary algebras with triangular decompositions.

A characterization of Reedy algebras using idempotent ideals in heredity chains is established.

The homological and representation-theoretic structures of Reedy algebras are elucidated.

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.