The Ringel dual of the Auslander-Dlab-Ringel algebra

The ADR algebra RAof a finite-dimensional algebra Ais a quasihereditary algebra. In this paper we study the Ringel dual R(RA)of RA. We prove that R(RA)can be identified with (RAop)op, under certain ‘minimal’ regularity conditions for A. In particular, over algebraically closed fields the Ringel dual of the ADR algebra RAis Morita equivalent to (RAop)op(with respect to a canonical labelling) if and only if all projective and injective A-modules are rigid and have the same Loewy length. We also give necessary and sufficient conditions for the ADR algebra to be Ringel selfdual.