On Constructions of Some Classes of Quasi-hereditary Algebras

Inspired by the work of Mirollo and Vilonen [MV] describing the categories of perverse sheaves as module categories over certain finite dimensional algebras, Dlab and Ringel introduced [DR2] an explicit recursive construction of these algebras in terms of the algebras A(γ). In particular, they characterized the quasi-hereditary algebras of Cline-Parshall-Scott [PS] and constructed them in this way. The present paper provides a characterization of lean algebras and some other special classes of algebras in terms of this recursive process. 1. Basics and Notation The aim of this paper is to clarify the structure of particular types of quasi-hereditary algebras which appear in the applications to the theory of Lie algebras. Our method is based on the construction described in [DR2], taking into account some characteristic properties of the bimodules and maps which define the recursive process to build up quasi-hereditary algebras of certain type. Let A be a basic finite dimensional K-algebra and e = eA = (e1, e2, e3, . . . , en) a complete sequence of primitive orthogonal idempotents of the algebra A so that ∑n i=1 ei = 1 : AA = n ⊕ i=1 eiA. Write εi = ei+ei+1+. . .+en for 1 ≤ i ≤ n and εn+1 = 0. Let us recall the definition of the right and left standard modules of A: ∆(i) = ∆A(i) = eiA / eiAεi+1A and ∆(i) = ∆A(i) = Aei / Aεi+1Aei, Research partially supported by Hungarian NFSR grant No. TO16432.