The Ringel dual of the Auslander-Dlab-Ringel algebra

TL;DR

This paper investigates the Ringel dual of the ADR algebra, establishing its relation to the opposite algebra and providing conditions for self-duality, thereby advancing understanding of quasihereditary algebra structures.

Contribution

It proves that the Ringel dual of the ADR algebra can be identified with the opposite of the ADR algebra of the opposite algebra, under specific conditions.

Findings

Ringel dual of ADR algebra is isomorphic to the opposite of the ADR algebra of the opposite algebra.

Provides necessary and sufficient conditions for ADR algebra to be Ringel selfdual.

Establishes a connection between the structure of ADR algebras and their duals.

Abstract

The ADR algebra $R_A$ of a finite-dimensional algebra $A$ is a quasihereditary algebra. In this paper we study the Ringel dual $\mathcal{R}(R_A)$ of $R_A$. We prove that $\mathcal{R}(R_A)$ can be identified with $(R_{A^{op}})^{op}$, under certain 'minimal' regularity conditions for $A$. We also give necessary and sufficient conditions for the ADR algebra to be Ringel selfdual.