All exact Borel subalgebras and all directed bocses are normal

TL;DR

This paper proves that all exact Borel subalgebras are inherently normal and that all directed bocses possess a group-like element, simplifying the existing bijection between these structures and quasihereditary algebras.

Contribution

It establishes that every exact Borel subalgebra is automatically normal, and every directed bocs has a group-like element, thereby simplifying previous bijections.

Findings

All exact Borel subalgebras are normal.

Every directed bocs has a group-like element.

Simplification of the bijection between bocses and quasihereditary algebras.

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.