スピン角運動量
S p i n ( 3 ) → π U ( V s ) ↷ V s Φ 3 ↓ S O ( 3 ) ↷ R 3 {\displaystyle {\begin{array}{rl}\mathrm {Spin} (3)&{\xrightarrow {~~~~~\pi ~~~~~}}{}\mathrm {U} (V_{s}){}^{\curvearrowright }V_{s}\\\Phi _{3}\downarrow &\\SO(3)&{}^{\curvearrowright }\mathbf {R} ^{3}\end{array}}}
πs が誘導する写像 (πs)*を以下のように定義する:
π ∗ : d  U ( t ) d  t | t = 0 ∈ s p i n ( 3 ) = s u ( 2 ) ↦ d  π s ( U ( t ) ) d  t | t = 0 ∈ { V s {\displaystyle \pi _{*}~:~\left.{\operatorname {d} U(t) \over \operatorname {d} t}\right|_{t=0}\in {\mathsf {spin}}(3)={\mathsf {su}}(2)\mapsto \left.{\operatorname {d} \pi _{s}(U(t)) \over \operatorname {d} t}\right|_{t=0}\in \{V_{s}} 上のエルミート演算子 } {\displaystyle \}} .