Triangular decompositions: Reedy algebras and quasi-hereditary algebras

TL;DR

This paper characterises finite-dimensional Reedy algebras as quasi-hereditary algebras admitting a triangular decomposition into two oppositely directed subalgebras over a common semisimple subalgebra, connecting Reedy structure to exact Borel and Delta subalgebras. It proves a key equivalence: Reedy algebras correspond to triples (A, B, C) with A ≅ C ⊗_S B and B, C quasi-hereditary with the same weight order, where B is an exact Borel subalgebra and C a Delta subalgebra. A second main result provides a recursive, idempotent-based characterisation: A is Reedy if and only if appropriate quotients and subalgebras A/AeA and eAe retain Reedy structure and the multiplication maps C ⊗_S B → A induce the required isomorphisms, enabling constructive assembly from smaller pieces. The work connects Reedy theory to the well-developed quasi-hereditary framework, yielding new examples (including monomial algebras and tensor products) and tools for transferring results between the two settings.

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 Poincaré-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.

Theorems & Definitions (27)

• Definition 3.1
• Example 3.2
• Example 3.3
• Example 3.4
• Example 3.5
• Lemma 3.6
• proof
• Remark 3.7
• Proposition 3.8
• proof