All quasihereditary algebras with a regular exact Borel subalgebra

Abstract

Not every quasihereditary algebra (A,Φ,⊴) has an exact Borel subalgebra. A theorem by Koenig, Külshammer and Ovsienko asserts that there always exists a quasihereditary algebra Morita equivalent to A that has a regular exact Borel subalgebra, but a characterisation of such a Morita representative is not directly obtainable from their work. This paper gives a criterion to decide whether a quasihereditary algebra contains a regular exact Borel subalgebra and provides a method to compute all the representatives of A that have a regular exact Borel subalgebra. It is shown that the Cartan matrix of a regular exact Borel subalgebra of a quasihereditary algebra (A,Φ,⊴) only depends on the composition factors of the standard and costandard A-modules and on the dimension of the Hom-spaces between standard A-modules. We also characterise the basic quasihereditary algebras that admit a regular exact Borel subalgebra.