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

研究了中心单代数到射影簇的函数域上的指数约简公式。所考虑的簇的类型是G/P的扭曲形式，其中G是分裂约化代数群，P是G的抛物子群。指数约简公式将根据感兴趣的簇的Grothendieck群的知识来推导，通常只有当G是连通的时才计算它。当G不连通时，首席研究者将研究G/P扭曲形式的Grothendieck群，从例子中可以清楚地看到，在连通情况下会出现一些新的现象。主要研究者还将研究具有对合项的中心单代数，以及相关的广义偶Clifford代数。本研究属于环论的一般领域。环是定义了加法和乘法的代数对象。虽然加法运算满足交换律，但乘法运算不是必须的。乘法不可交换的环的一个例子是整数上的n×n矩阵的集合。非对易环的研究已经成为代数的重要组成部分，因为它对数学和物理的其他分支的意义越来越大。

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.