Spin(n) (n ≤ 5)的矩阵表示

The Matrix Representations of Spin(n) for n ≤ 5

摘要: 本文基于四元数，对低维自旋群Spin(n) (n=2,3,4,5)给出了实与复矩阵表示的显式构造。 我们 从Clifford 代数出发，系统地导出了这些群的矩阵表示，并利用经典群表示直接给出了相应的旋 量表示。 整个过程清楚地展示了例外同构 Spin(2) ∼= U(1), Spin(3) ∼= SU(2), Spin(4) ∼= SU(2) × SU(2), Spin(5) ∼= Sp(2), 并建立了矩阵表示与上述同构之间的具体对应。 这些结果可直接应用于几何、 物理和计算机图形 学等领域。

Abstract: Based on quaternions, this paper presents explicit constructions of real and complex matrix representations for the spin groups Spin(n) for n= 2, 3, 4, 5. Starting from the Clifford algebra formalism, we compute both the matrix representations and the spinor representations of Spin(n), with the spinor representations being directly ob- tained from those of the classical groups. The treatment highlights the well-known exceptional isomorphisms Spin(2) ∼= U(1), Spin(3) ∼= SU(2), Spin(4) ∼= SU(2) × SU(2), Spin(5) ∼= Sp(2), and characterizes the concrete connection between the matrix representations and these isomorphisms. These results can be directly applied to geometry, physics, and computer graphics.