Tuesday, October 27, 2020

Teresa Conde, Quasihereditary algebras with exact Borel subalgebras

Abstract:
Not every quasihereditary algebra A 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 Morita representative is not directly obtainable from their work. In this talk, a criterion to decide whether a quasihereditary algebra contains a regular exact Borel subalgebra shall be introduced and a method to compute all Morita representatives of A that have a regular exact Borel subalgebra will be presented. 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. We conclude the talk with a characterisation of the basic quasihereditary algebras that admit a regular exact Borel subalgebra.