Tits'systems in Chevalley groups over Laurent polynomial rings
Tits'systems in Chevalley groups over Laurent polynomial rings
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洛朗多项式环上 Chevalley 群中的 Tits 系统
DOI：
10.21099/tkbjm/1496158687
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发表时间：
1979
影响因子：
0.7
通讯作者：
Jun Morita
中科院分区：
文献类型：
--
作者：
Jun Morita
本文的目的是证明Laurent多项式环中Chevailey群的初等子群具有一个包含一个Laurent Weyl群的Tits系统的结构（关于Tits系统，见[2]）.我们让Z表示有理整数。设A是一个（约化的）根系（cf. [2]，[4]）。则存在一个有限维复半单李代数L = L（A），它的根系为J，且在同构下唯一。设G是一个与L和p相关联的Chevalley-Demazure群概型（定义见[1]，[8]）.由于G是从具有1的交换环范畴到群范畴的可表示协变函子，因此我们得到了具有1的交换环R的点的群G（R）。我们称G（R）为R上的Chevailey群。对于每个根a ∈ A，存在R的加法群R* 到G（R）的子群Xa上的群同构（参见图1）。[1]，[8]）。初等子群E（R）被定义为G（R）的由X_a生成的子群。如果A是AL型，p是泛型（参见[4])，则G（R）=SLi+i（R），E（R）是SXm（J？）由/J+i+fley生成，用于所有对象i？其中It+1是（i +1）x（i +1）单位矩阵，并且ej是矩阵单位（i，j位置为1，其他位置为0）。如果是2？是域，则E（R）具有与A的Weyl群相关联的Tits'系统的结构（cf. [9]）。如果R是一个具有离散赋值的域，则E（R）具有与A的仿射Weyl群相关联的Tits'系统的结构（参见：[5]）。设K[T，71 - 1]是T和T~ 1中的Laurent多项式环，其系数在域K中.本文证明了E{K[T，T~r\）具有与A的仿射Weyl群相关联的Tits系统的结构。设Lz是L中的Chevailey格（参见[4])并且设置%K=K[T，T^y&zLz。则%K同构于一个欧几里得李代数（参见[6]）。因此，如果p是伴随类型，并且如果
Our aim is to show that the elementary subgroup of a Chevailey group ovei a Laurent polynomial ring has the structure of a Tits' system with an affim Weyl group (as for Tits' system, see [2]). We let denote Z the rational integers. Let A be a (reduced) root system (cf.[2],[4]). Then there is a finite dimensional complex semisimple Lie algebra L = L(A), unique up to isomorphism, whose root system is J. Let p be a finitedimensional complex faithfulrepresentation of L. Let G be a Chevalley-Demazure group scheme associated with L and p (as for the definition,see [1],[8]). Since G is a representable covariant functor from the category of commutative rings with 1 to the category of groups, we get a group G(R) of the points of a commutative ring R, with 1. We call G(R) a Chevailey group over R. For each root a£A, there is a group isomorphism of the additive group R* of R onto a subgroup Xa of G(R) (cf.[1],[8]). The elementary subgroup E(R) is defined to be the subgroup of G(R) generated by Xa for all≪eJ. If A is of type AL and p is of universal type (cf.[4]),then G(R)=SLi+i(R) and E(R) is the subgroup EU1(R) of SXm(J?) generated by /J+i+fley for all ≪ei? and l<i'^j<l+l, where It+1 is the (/+l)x(/+l) identity matrix and e^ is a matrix unit (1 in the i,j position,0 elsewhere). If 2? is a field,then E(R) has the structure of a Tits' system associated with the Weyl group of A (cf.[9]). If R is a fieldwith a discretevaluation, then E(R) has the structure of a Tits' system associated with the affine Weyl group of A (cf.[5]). Let K[T, 71"1]be the ring of Laurent polynomials in T and T~l with coefficientsin a fieldK. In this paper, we will show that E{K[T, T~r\) has the structure of a Tits' system associated with the affine Weyl group of A. Let Lz be a Chevailey latticein L (cf.[4]) and set %K=K[T, T^y&zLz. Then %K is isomorphic to a Euclidean Lie algebla (cf.[6]). Thus, if p is of adjoint type, and if