群上同调在Fourier空间准晶分类中的应用
Applications of Group Cohomology to the Classification of Fourier-Space Quasicrystals
摘要：
在1962年，Bienenstock和Ewald在对偶空间或傅立叶空间中用代数描述了晶体空间群的分类。最近，该方法已被应用于准晶和调制晶体。本文用群上同调的概念来表述Bienenstock和Ewald的定义。定义了一个\textit{Fourier quasicrystal}及其空间群，而不要求它来自实（直接）空间中的准晶。某个上同调群分类了与给定点群和拟格相关的空间群，对偶同调群给出了所有的规范不变量。利用这种对偶性来证明以前只在特殊情况下才知道的几个结果，包括二维任意秩准格的空间群的分类。X射线衍射图中的消光和电子能级的简并被解释为非零同源类的物理表现。
英文摘要：
In 1962, Bienenstock and Ewald described the classification of crystalline space groups algebraically in the dual, or Fourier, space. Recently, the method has been applied to quasicrystals and modulated crystals. This paper phrases Bienenstock and Ewald's definitions in terms of group cohomology. A \textit{Fourier quasicrystal} is defined, along with its space group, without requiring that it come from a quasicrystal in real (direct) space. A certain cohomology group classifies the space groups associated to a given point group and quasilattice, and the dual homology group gives all gauge invariants. This duality is exploited to prove several results that were previously known only in special cases, including the classification of space groups for quasilattices of arbitrary rank in two dimensions. Extinctions in X-ray diffraction patterns and degeneracy of electronic levels are interpreted as physical manifestations of non-zero homology classes.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
