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Exact Borel subalgebras of quasihereditary algebras emulate the role of "classic" Borel subalgebras of complex semisimple Lie algebras. Not every quasihereditary algebra A has an exact Borel subalgebra. However, a theorem by Koenig, Külshammer and Ovsienko establishes that there always exists a quasihereditary algebra Morita equivalent to A that has a (regular) exact Borel subalgebra. Despite that, a characterisation of such "special" Morita representatives is not directly obtainable from Koenig, Külshammer and Ovsienko's work. In this talk, I shall present a numerical criterion to decide whether a quasihereditary algebra contains a regular exact Borel subalgebra and I will provide a method to compute all Morita representatives of A that have a regular exact Borel subalgebra. We shall also see that the Cartan matrix of a regular exact Borel subalgebra of a quasihereditary algebra A 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. I will conclude the talk with a characterisation of the basic quasihereditary algebras that admit a regular exact Borel subalgebra.
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