This note presents an elementary iterative construction of the generators for the complex $K$-groups $K_i\left(C\left({\mathbb{S}}^d\right)\right)$ of the $d$-dimensional spheres. These generators are explicitly given as the restrictions of Dirac or Weyl Hamiltonians to the unit sphere. Connections to solid state physics are briefly elaborated on.

中文翻译：

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球体 K 群的生成元

本说明介绍了复杂的生成器的基本迭代结构$K$-组$K_{我}（C（{\mathbb{S}}^d））$的$d$维度球体。这些生成元被明确给出为狄拉克或韦尔哈密顿量对单位球体的限制。简要阐述了与固态物理学的联系。