Some tits systems with affine Weyl groups in Chevalley groups over Dedekind domains
Some tits systems with affine Weyl groups in Chevalley groups over Dedekind domains
复制标题
Dedekind 域上 Chevalley 群中具有仿射 Weyl 群的一些山雀系统
DOI：
10.1016/0021-8693(88)90272-4
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发表时间：
1988
影响因子：
0.9
通讯作者：
Jun Morita
中科院分区：
文献类型：
--
作者：
E. Abe;Jun Morita
设A是Dedekind整环，P是A的非零素理想，使得P=（p）是主理想，乘法单位群的标准同态Ax+（A/P）x是满射.设R是A的商域的子环A [;]。本文对R上的Chevalley群给出了具有仿射Weyl群的Tits系统的结构，并给出了这种群结构的一些应用.我们的结果包含作为特殊情况的结构的Tits系统的仿射Weyl群的Chevalley群在一个域上的非平凡的离散赋值由于N。Iwahori和H.松本[141]和J. Morita [171.本文将Laurent多项式环上的Chevalley群看作是与仿射型Kac-Moody李代数相联系的群，并利用D. Peterson和V. Kac [191]对与Kac-Moody李代数相关的群的研究。由于我们不一定把Dedekind整环上的群当作域或局部环来处理，所以这个结果有一些应用。我们得到了一些Dedekind域是泛域或拟泛域的一个充分条件，并且得到了R上Chevalley群的一种Iwasawa分解。
Let A be a Dedekind domain and P be a nonzero prime ideal of A such that P=(p) is principal and the canonical homomorphism A x+(A/P) x of multiplicative groups of units is surjective. Let R be the subring A [;] of the quotient field of A. In this note, we give for a Chevalley group over R a structure of a Tits system with the affine Weyl group and give some applications of such a structure of the groups. Our result contains as special cases the structure of Tits systems with affine Weyl groups in a Chevalley group over a field with a non-trivial discrete valuation due to N. Iwahori and H. Matsumoto [141 and in a Chevalley group over a Laurent polynomial ring due to J. Morita [171. The Chevalley groups over a Laurent polynomial ring are regarded as groups associated with Kac-Moody Lie algebras of affine types, and our proof of the theorem is based on a technique used by D. Peterson and V. Kac [191 for groups associated with Kac-Moody Lie algebras. Since we treat the groups over Dedekind domains not necessarily as fields or local rings, there are some applications of the result. We can obtain a sufficient condition for some Dedekind domains to be universal or quasi-universal, and also we have a sort of Iwasawa decomposition of Chevalley groups over R.