The Ringel dual of the Auslander-Dlab-Ringel algebra

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.