On certain strongly quasihereditary algebras


Given a finite-dimensional algebra $A$ we may associate to it a special endomorphism algebra, $R_A$, introduced by Auslander. The algebra $R_A$ is a ‘Schur-like’ algebra for $A$: it contains $A$ as an idempotent subalgebra (up to Morita equivalence) and it is quasihereditary with respect to a particular heredity chain. The main purpose of this thesis is to describe the quasihereditary structure of $R_A$ which arises from such heredity chain, and to investigate the corresponding Ringel dual of $R_A$. It turns out that $R_A$ belongs to a certain class of strongly quasihereditary algebras defined axiomatically, which we call ultra strongly quasihereditary algebras. We derive the key properties of ultra strongly quasihereditary algebras, and give examples of other algebras which fit into this setting.

PhD thesis, University of Oxford