All quasihereditary algebras with a regular exact Borel subalgebra

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