Group actions on C*-algebras, 3-cocycles and quantum field theory
Group actions on C*-algebras, 3-cocycles and quantum field theory
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C* 代数、3-余循环和量子场论的群作用
DOI：
10.1007/bf02101555
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发表时间：
1995
影响因子：
2.4
通讯作者：
C. Sutherland
中科院分区：
文献类型：
--
作者：
A. L. Carey;H. Grundling;I. Raeburn;C. Sutherland
我们研究群扩张Δ→Γ→Ω，其中Γ作用在C*-代数A上。给定对A，Δ的扭曲协变表示π，V，我们构造Ω上的3-上圈，其值在V（Δ）生成的群的中心.这些3-上圈是存在Ω by V（Δ）的扩张的障碍，该扩张与Γ相容地作用在A上.本文的主要定理引入了一个辅助不变量Λ，它对Γ onV（Δ）的作用进行分类，并由此给出了3-上圈的上同调类为非平凡的充要条件。提供的例子表明如何非平凡的3-cocycles可能实现。我们选择展示这些基本数学结果的框架受到反常规范场理论的影响。我们将展示如何解释我们的结果，在这种情况下，在两种方式，一个动机的代数方法约束动力学和其他下降方程的方法来构建规范群上的cocycles。为了与规范理论中通常的上同调方法进行比较，我们得出了不变量Λ和3-上圈的李代数形式。
We study group extensions Δ→Γ→Ω, where Γ acts on a C*-algebraA. Given a twisted covariant representation π,V of the pairA, Δ we construct 3-cocycles on Ω with values in the centre of the group generated byV(Δ). These 3-cocycles are obstructions to the existence of an extension of Ω byV(Δ) which acts onA compatibly with Γ. The main theorems of the paper introduce a subsidiary invariant Λ which classifies actions of Γ onV(Δ) and in terms of which a necessary and sufficient condition for the the cohomology class of the 3-cocycle to be non-trivial may be formulated. Examples are provided which show how non-trivial 3-cocycles may be realised. The framework we choose to exhibit these essentially mathematical results is influenced by anomalous gauge field theories. We show how to interpret our results in that setting in two ways, one motivated by an algebraic approach to constrained dynamics and the other by the descent equation approach to constructing cocycles on gauge groups. In order to make comparisons with the usual approach to cohomology in gauge theory we conclude with a Lie algebra version of the invariant Λ and the 3-cocycle.