Recently, Brzeziński, Koenig and Külshammer 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 Brzeziński, Koenig and Külshammer's bijection.

中文翻译：

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所有精确的Borel子代数和所有有向Bocses都是正常的

最近，布热津​​斯基（Brzeziński），科尼希（Koenig）和库尔瑟默（Külshammer）引入了准遗传代数的正态精确Borel次代数的概念。他们表明，在正常有向Bocses和准正态代数与正则和同源精确Borel次代数之间存在一对一的对应关系。在此简短说明中，我们证明了每个精确的Borel子代数都是自动正态的。作为推论，我们得出结论，每个定向bocs都有一个类似组的元素。这些结果简化了Brzeziński，Koenig和Külshammer的双射。