利用分次子代数研究表示论中的Koszul性质
A new approach to the Koszul property in representation theory using graded subalgebras
摘要：
给定一个拟遗传代数$B$，给出了保证将$B$按根滤除分阶得到的代数$\gr B$是Koszul的条件，同时也保证了$B$可能具有的拟遗传性质和其它良好的李论性质.该方法涉及到一对$(A,{\mathfrak a})$组成的拟遗传代数$A$和（正）分次子代数$\mathfrak a$。代数$B$由$A$的定义理想$J$作为$A$的商$B=A/J$而产生。一路上，我们还表明，$B$的标准（Weyl）模块有一个结构作为分级模块$\mathfrak a$。这些结果被应用于获得关于有限维代数的新信息（例如，$q$ -Schur代数），它们是量子包络代数的子代数。进一步的应用，也许是最具穿透力的，产生结果的有限维代数相关联的半单代数群的积极特征$p$。这些结果要求，至少目前，对$p$的大小进行相当大的限制。
英文摘要：
Given a quasi-hereditary algebra $B$, we present conditions which guarantee that the algebra $\gr B$ obtained by grading $B$ by its radical filtration is Koszul and at the same time inherits the quasi-hereditary property and other good Lie-theoretic properties that $B$ might possess. The method involves working with a pair $(A,{\mathfrak a})$ consisting of a quasi-hereditary algebra $A$ and a (positively) graded subalgebra $\mathfrak a$. The algebra $B$ arises as a quotient $B=A/J$ of $A$ by a defining ideal $J$ of $A$. Along the way, we also show that the standard (Weyl) modules for $B$ have a structure as graded modules for $\mathfrak a$. These results are applied to obtain new information about the finite dimensional algebras (e.g., the $q$-Schur algebras) which arise as quotients of quantum enveloping algebras. Further applications, perhaps the most penetrating, yield results for the finite dimensional algebras associated to semisimple algebraic groups in positive characteristic $p$. These results require, at least presently, considerable restrictions on the size of $p$.
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