Mathematical Sciences: Index Reduction Formulas for Central Simple Alge>ras.

本文研究了中心简单代数在射影变函数域的通道上的指数约简公式。所考虑的变量类型是G/P的一个扭曲形式，其中G是一个分裂约化代数群，P是G的一个抛物子群。指数约化公式将从所关注的变量的Grothendieck群的知识推导出来，它一般只在G连通时才计算出来。当G不连通时，首席研究员将研究G/P的扭曲形式的Grothendieck群，从例子中可以清楚地看出，一些在连通情况下没有看到的新现象将出现。主要研究者还将研究有对合的中心简单代数，以及相关的广义偶数Clifford代数。本研究属于环理论的一般范畴。环是一个代数对象，在它上面定义了加法和乘法。虽然加法运算满足交换律，但乘法运算不需要满足交换律。一个乘法不可交换的环的例子是整数上的nxn个矩阵的集合。非交换环的研究由于其对其他数学和物理分支的重要意义而成为代数的重要组成部分。

This work is concerned with the study of index reduction formulas for central simple algebras on passage to the function field of a projective variety. The type of variety under consideration is a twisted form of G/P where G is a split reductive algebraic group and P is a parabolic subgroup of G. The index reduction formula will follow from a knowledge of the Grothendieck group of the variety of interest, which has been computed in general only when G is connected. The principal investigator will investigate the Grothendieck group of the twisted forms of G/P when G is not connected, where it is clear from examples that some new phenomena not seen in the connected case will arise. The principal investigator will also study central simple algebras with involution, and the associated generalized even Clifford algebras. This research is in the general area of ring theory. A ring is an algebraic object having both an addition and a multiplication defined on it. Although the additive operation satisfies the commutative law, the multiplicative operation is not required to do so. An example of a ring for which multiplication is not commutative is the collection of nxn matrices over the integers. The study of noncommutative rings has become an important part of algebra because of its increasing significance to other branches of mathematics and physics.