Ringel duality for certain strongly quasi-hereditary algebras

Abstract

We study quasi-hereditary endomorphism algebras defined over a new class of finite dimensional monomial algebras with a special ideal structure. The main result is a uniform formula describing the Ringel duals of these quasi-hereditary algebras.As special cases, we obtain a Ringel-duality formula for a family of strongly quasihereditary algebras arising from a type A configuration of projective lines in a rational, projective surface as recently introduced by Hille and Ploog, for certain Auslander-Dlab-Ringel algebras, and for Eiriksson and Sauter's nilpotent quiver algebras when the quiver has no sinks and no sources. We also recover Tan's result that the Auslander algebras of self-injective Nakayama algebras are Ringel self-dual. Ringel duality [Rin91] is a fundamental phenomenon in the theory of quasi-hereditary algebras, see for example [Kra13,GGOR03,BK17,CM17,Puč15,IR11,EP04,CE17,Cou17] for (recent) work on this topic. For any quasi-hereditary algebra A there exists a quasihereditary algebra R(A), the Ringel-dual of A, such that A-mod ∼ = R(R(A))-mod.

Characterizations of right rejective chains

“…Quasi-hereditary algebras were introduced by Cline, Parshall, and Scott to explore highest weight categories arising from the study of semisimple complex Lie algebras and algebraic groups [5,20]. As a distinguished class of quasi-hereditary algebras, Ringel [18] introduced right-strongly quasi-hereditary algebras which frequently appear in the representation theory of algebras [7,8,10,16,21]. It is known that special right rejective chains called total right rejective chains, are characterized by right-strongly quasi-hereditary algebras.…”