Prime spectra of quantum semisimple groups
Prime spectra of quantum semisimple groups
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量子半单群的素谱
DOI：
10.1090/s0002-9947-96-01597-8
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发表时间：
1996
影响因子：
1.3
通讯作者：
K. Goodearl
中科院分区：
文献类型：
--
作者：
K. Brown;K. Goodearl
本文研究了量子化函数代数Rq [G]的素理想空间，其中G是半单李群，q是不定的.我们的方法是检查满足一组七个假设的代数的结构，然后使用Joseph，Hodges和Levasseur的工作来证明代数Rq [G]满足这一系列假设。满足这些假设的环被证明满足正规分离性，因此满足Jategaonkar的强第二层条件。对于这样的环表示理论的信息是由图的链接的素谱，所以我们进行了详细的研究素链接的代数满足清单的假设。齐性是一个重要特征，证明了在有限秩自由交换自同构群下，任何素理想的团与其轨道重合。在Rq [G]的情形下，得到了这些群秩的界.在最后一节的结果是专门的情况G = SLn（C），详细的计算可以用来说明一般的结果。作为一组初步的例子，我们还表明，多参数量子坐标环的仿射n-空间满足我们的公理方案时，由参数生成的组是torsionfree的。0.导言和背景0.1.设P和Q是Noether环R的素理想。我们称P与Q相连，记为P -*Q，如果存在R的理想A，且PQCAcPnQ使得（PnQ）/A是非零挠自由左（R/P）-模和挠自由右（R/Q）-模。在R满足第二层条件的情况下，该条件可以被简化（见下文）。在这种情况下，P -* Q当且仅当（P n Q）/PQ作为左（R/P）-模和右（R/Q）-模是忠实的[39; 10，1.4]。R的链图是顶点为规格R（R的素理想集）的点的有向图，有从P到Q的有向边当且仅当P -* Q。包含P的链环图的连通分支中的素数构成P的团，记为团（P）。R的环图和R的表示论之间有着密切的联系（参见图1）。[20]，[4]）;当R满足强第二层条件时，这种联系特别强，该条件要求，无论何时U是具有零化子的循环一致R-模。1991年数学学科分类。初级16 D30、16 D 60、16 P40、17 B37。第二作者的研究部分得到了美国国家科学基金会的资助。部分工作进行了，而他访问了格拉斯哥大学数学系在1993年10月，支持爱丁堡和伦敦数学学会。工作的修订版的文件是在1995年夏天完成的访问期间，两位作者的数学系的华盛顿大学，谁都感谢其款待。第一提交人的旅费部分由卡内基信托基金为苏格兰大学提供的赠款支付。(?) 1996美国数学学会
We study the prime ideal spaces of the quantized function algebras Rq [G], for G a semisimple Lie group and q an indeterminate. Our method is to examine the structure of algebras satisfying a set of seven hypotheses, and then to demonstrate, using work of Joseph, Hodges and Levasseur, that the algebras Rq [G] satisfy this list of assumptions. Rings satisfying the assumptions are shown to satisfy normal separation, and therefore Jategaonkar's strong second layer condition. For such rings much representation-theoretic information is carried by the graph of links of the prime spectrum, and so we proceed to a detailed study of the prime links of algebras satisfying the list of assumptions. Homogeneity is a key feature it is proved that the clique of any prime ideal coincides with its orbit under a finite rank free abelian group of automorphisms. Bounds on the ranks of these groups are obtained in the case of Rq [G]. In the final section the results are specialized to the case G = SLn (C), where detailed calculations can be used to illustrate the general results. As a preliminary set of examples we show also that the multiparameter quantum coordinate rings of affine n-space satisfy our axiom scheme when the group generated by the parameters is torsionfree. 0. INTRODUCTION AND BACKGROUND 0.1. Let P and Q be prime ideals of a noetherian ring R. We say that P is linked to Q, and write P -*Q, if there is an ideal A of R with PQ C A c P n Q such that (P n Q)/A is a nonzero torsionfree left (R/P)-module and a torsionfree right (R/Q)-module. This condition can be simplified in case R satisfies the second layer condition (see below). In that case, P -* Q if and only if (P n Q)/PQ is faithful as a left (R/P)-module and as a right (R/Q)-module [39; 10, 1.4]. The graph of links of R is the directed graph whose vertices are the points of spec R (the set of prime ideals of R), with a directed edge from P to Q if and only if P -* Q. The primes in the connected component of the graph of links containing P constitute the clique of P, denoted clique(P). There is a close connection between the graph of links of R and the representation theory of R (cf. [20], [4]); this connection is particularly strong when R satisfies the strong second layer condition, which requires that, whenever U is a cyclic uniform R-module with the annihilator Received by the editors November 4, 1994 and, in revised form, September 5, 1995. 1991 Mathematics Subject Classification. Primary 16D30, 16D60, 16P40, 17B37. The research of the second author was partially supported by a grant from the National Science Foundation (USA). Part of the work was carried out while he visited the University of Glasgow Mathematics Department during October 1993, supported by the Edinburgh and London Mathematical Societies. Work on a revised version of the paper was completed in summer 1995 during a visit by both authors to the Department of Mathematics of the University of Washington, whom both thank for its hospitality. The travel costs of the first author were in part covered by a grant from the Carnegie Trust for the Universities of Scotland. (?)1996 American Mathematical Society