关于量子一般线性群的几何点的一个刻划(英文)

On a Characterization of Geometric Points of Quantum General Linear Groups

按[1]的观点，我们把量子群看成广义的仿射群概形。若R∈Ob（AlgK），则集合GLn，q（R）由满足如下两个条件的矩阵α＝（αij）n×n∈Mn（R）组成（这里Mn（R）是R上所有n×n矩阵的集合）：（1）αij满足一组众所周知的关系式（参看（1．1））；（2）Dq（α）在R中可逆，这里Dq为量子行列式（参看（1．2））。本文证明了条件（2）等价于如下条件：（3）α在Mn（R）中可逆。Cartier在论文[3]中用条件（1）和（3）来定义集合GLn，q（R）。因此，本文的结果保证了前面定义的GLn，q（R）与Cartier[3]所定义的GLn，q（R）是一致的。

Following [1], we regard a quantum group as a generalized affine group scheme.For R ∈Ob (AlgK), GLn. q(R ) consists of the matrics a = (aij )n ×n ∈ Mn (R ) (the set of n × n - matrices with entries in R ) with the following two properties: (1) aij s satisfy the wen - known defining reations (see (1. 1 ) );(2) Dq(a ) is invertible in R, Dq being the quantum determinant (see (1. 2)). In this paper we proved that conditha (2) is equlvalent to the following condition;(3) a is invertible in Mn(R ).Note that conditiona (1) and (3) are used by Cartier[3] to define the set GLn, q (R ). Thus,our result ensures that the above definition of GLn,q (R ) coincides with the definition given by Carties[3].