All quasihereditary algebras with aregular exact Borel subalgebra

Not every quasihereditary algebra (A, Phi, (sic)) has an exact Borel subalgebra. A theorem by Koenig, Kulshammer 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, Phi, (sic)) 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. (C) 2021 Elsevier Inc. All rights reserved.

Automated summary

This paper provides a criterion to determine whether a quasihereditary algebra contains a regular exact Borel subalgebra and presents a method to compute all representatives of A that have a regular exact Borel subalgebra. It is also shown that the Cartan matrix of a regular exact Borel subalgebra depends only on the composition factors of standard and costandard A-modules and the dimension of the Hom-spaces between standard A-modules. Additionally, the paper characterises the basic quasihereditary algebras that admit a regular exact Borel subalgebra.