论文标题
统一量子群集代数和判别因子的根
Root of unity quantum cluster algebras and discriminants
论文作者
论文摘要
我们描述了群集代数，多项式身份代数和判别因子的主题之间的联系。为此，我们定义了Unity量子群集代数的词根的概念，并证明它们是多项式认同代数。在每个这样的代数内部，我们构建了一个（大）规范的中央亚词法，可以看作是De Concini，kac和procesi构建的大量子基团的中央子代数的远程概括，并在表示理论中使用。事实证明，每种这种中央亚词法都是与几何类型的基本经典群集代数同构的。当Unity量子群集代数的根部在其中央亚词法上是自由的时，我们证明了这对的判别物是冷冻变量的力量的产物。对于由Exchange图中神经的簇变量产生的所有亚代词的判别物也证明了该结果的扩展。这些结果可用于有效计算判别因素。作为一个应用程序，我们证明了$ {\ mathbb {z}}超过$ {\ mathbb {z}} [\ varepsilon] $ de concini，kac和procesi的每个量子单元单元的明确公式，用于任意同性恋的kac-moody elgebras，其中$ \ varepsilon $ is unity是一定的。
We describe a connection between the subjects of cluster algebras, polynomial identity algebras and discriminants. For this, we define the notion of root of unity quantum cluster algebras and prove that they are polynomial identity algebras. Inside each such algebra we construct a (large) canonical central subalgebra, which can be viewed as a far reaching generalization of the central subalgebras of big quantum groups constructed by De Concini, Kac and Procesi and used in representation theory. Each such central subalgebra is proved to be isomorphic to the underlying classical cluster algebra of geometric type. When the root of unity quantum cluster algebra is free over its central subalgebra, we prove that the discriminant of the pair is a product of powers of the frozen variables times an integer. An extension of this result is also proved for the discriminants of all subalgebras generated by the cluster variables of nerves in the exchange graph. These results can be used for the effective computation of discriminants. As an application we prove an explicit formula for the discriminant of the integral form over ${\mathbb{Z}}[\varepsilon]$ of each quantum unipotent cells of De Concini, Kac and Procesi for arbitrary symmetrizable Kac-Moody algebras, where $\varepsilon$ is a root of unity.
