论文标题
所有准精确的Borel子代数
All quasihereditary algebras with a regular exact Borel subalgebra
论文作者
论文摘要
并非每个准标准代数$（a，φ，\ unlhd）$都有一个精确的borel subergebra。 Koenig，Külshammer和Ovsienko的定理断言，始终存在一个准单位代数莫里塔（Morita），等同于$ a $ a $具有常规精确的Borel subgebra，但是从工作中无法直接获得这种莫里塔（Morita）代表的特征。本文给出了一个标准，以决定准植虫的代数是否包含定期精确的Borel subergebra，并提供了一种计算具有定期精确Borel Borel subgebra的$ A $代表的方法。结果表明，quasihereparity代数$（a，φ，\ linhd）$的常规确切borel borel subgerbra的cartan矩阵仅取决于标准和costandard $ a $ a $ - mmodules的组成因子，以及$ \ persatateNarnemmodules and $ \ persatateNorn operational {home} $ - pass $ - pass $ -odul a $ -odul。我们还表征了接受定期精确的Borel子代数的基本准植虫代数。
Not every quasihereditary algebra $(A,Φ,\unlhd)$ 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 $(A,Φ,\unlhd)$ only depends on the composition factors of the standard and costandard $A$-modules and on the dimension of the $\operatorname{Hom}$-spaces between standard $A$-modules. We also characterise the basic quasihereditary algebras that admit a regular exact Borel subalgebra.
