All quasihereditary algebras with a regular exact Borel subalgebra

Abstract

Not every quasihereditary algebra $(A,\Phi ,⊴)$ 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 $(A,\Phi ,⊴)$ only depends on the composition factors of the standard and costandard $A$-modules and on the dimension of the $\mathrm{Hom}$-spaces between standard $A$-modules. We also characterise the basic quasihereditary algebras that admit a regular exact Borel subalgebra.

39 pages; v2: signs corrected in Lemma 3.3, other typos corrected

References in corpus (4)

• A-infinity structures on the algebra of extensions of Verma modules in the parabolic category O
• Combinatorics of quasi-hereditary structures
• The classification of blocks in BGG category O
• All exact Borel subalgebras and all directed bocses are normal

Cited by in corpus (1)

• The $\mathrm{Ext}$-algebra of standard modules over dual extension algebras