All exact Borel subalgebras and all directed bocses are normal

Abstract

Recently, Brzezi\'nski, Koenig and K\"ulshammer have introduced the notion of normal exact Borel subalgebra of a quasihereditary algebra. They have shown that there exists a one-to-one correspondence between normal directed bocses and quasihereditary algebras with a normal and homological exact Borel subalgebra. In this short note, we prove that every exact Borel subalgebra is automatically normal. As a corollary, we conclude that every directed bocs has a group-like element. These results simplify Brzezi\'nski, Koenig and K\"ulshammer's bijection.