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 quasi-hereditary 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.
Highlights
Quasi-hereditary algebras form an important class of finite dimensional algebras with relations to Lie theory and exceptional sequences in algebraic geometry
We study quasi-hereditary endomorphism algebras defined over a new class of finite dimensional monomial algebras with a special ideal structure
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
Summary
Quasi-hereditary algebras form an important class of finite dimensional algebras with relations to Lie theory (this was the original motivation [10]) and exceptional sequences in algebraic geometry (see e.g. [9,23]). The Ringel duality formula of Theorem 1.2, the definition of ideally ordered monomial algebras, and the construction of the algebras ER in this paper are all geometrically inspired. They were first observed in our previous work [27] for a class of quasihereditary algebras α constructed by Hille and Ploog [24]. The aim of this paper was to find a more general representation theoretic framework extending the Ringel duality formula (1) to a larger class of (ultra) strongly quasi-hereditary algebras. Work of Dlab and Ringel [17] shows that every finite dimensional algebra admits a noncommutative ‘resolution’ by a quasi-hereditary algebra, and a generalisation of this result led to Iyama’s proof of the finiteness of Auslander’s representation dimension [25]. For the injective cogenerator I ..= R†R we define the category of divisible R-modules fac(I ) ..= add{Q | I ⊕n → Q → 0} ⊆ R-mod and let FAC(I ) ..= Q∈ind(fac(I )) Q denote the direct sum of all indecomposable objects in fac(I ) up to isomorphism
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