Borelic pairs for stratified algebras

Abstract

We determine all values of the parameters for which the cell modules form a standard system, for a class of cellular diagram algebras including partition, Brauer, walled Brauer, Temperley–Lieb and Jones algebras. For this, we develop and apply a general theory of finite dimensional algebras with Borelic pairs. The theory is also applied to give new uniform proofs of the cellular and quasi-hereditary properties of the diagram algebras and to construct quasi-hereditary 1-covers, in the sense of Rouquier, with exact Borel subalgebras, in the sense of König. Another application of the theory leads to a proof that Auslander–Dlab–Ringel algebras admit exact Borel subalgebras.