Root of unity quantum cluster algebras and Cayley–Hamilton algebras
Root of unity quantum cluster algebras and Cayley–Hamilton algebras
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单位根量子簇代数和凯莱汉密尔顿代数
DOI：
10.1090/tran/8904
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发表时间：
2023
影响因子：
1.3
通讯作者：
Yakimov, Milen
中科院分区：
文献类型：
--
作者：
Huang, Shengnan;Lê, Thang;Yakimov, Milen
我们证明了单位量子簇代数根框架中的大类代数具有Procesi意义上的中心简代数和Cayley-Hamilton代数的最大阶结构。我们证明了单位上量子簇代数的每个根都是最大阶，并得到了其约简迹的显式公式。在温和的假设下，在每个这样的代数内部，我们构造一个与底层上簇代数同构的规范中心子代数，使得该对是凯利-汉密尔顿代数；它的完整 Azumaya 基因座显示包含基础簇品种的副本。这两个结果都在混合量子环面在种子子集合上的交集的更广泛的通用性中得到了证明。此外，我们证明了单位量子环面根的所有单项式子代数都是凯莱-哈密尔顿代数，并对那些最大阶的子代数进行了分类。种子子集上的任意交集也被证明是凯莱-汉密尔顿代数。以前构建最大阶数的方法依赖于过滤和同源方法。我们使用基于簇代数的新方法。参考
We prove that large classes of algebras in the framework of root of unity quantum cluster algebras have the structures of maximal orders in central simple algebras and Cayley–Hamilton algebras in the sense of Procesi. We show that every root of unity upper quantum cluster algebra is a maximal order and obtain an explicit formula for its reduced trace. Under mild assumptions, inside each such algebra we construct a canonical central subalgebra isomorphic to the underlying upper cluster algebra, such that the pair is a Cayley–Hamilton algebra; its fully Azumaya locus is shown to contain a copy of the underlying cluster-variety. Both results are proved in the wider generality of intersections of mixed quantum tori over subcollections of seeds. Furthermore, we prove that all monomial subalgebras of root of unity quantum tori are Cayley–Hamilton algebras and classify those ones that are maximal orders. Arbitrary intersections of those over subsets of seeds are also proved to be Cayley–Hamilton algebras. Previous approaches to constructing maximal orders relied on filtration and homological methods. We use new methods based on cluster algebras. References