# 旋量群

${\displaystyle 1\to \mathbb {Z} _{2}\to \operatorname {Spin} (n)\to \operatorname {SO} (n)\to 1}$

n > 2， Spin(n) 单连通，从而是 SO(n) 的万有覆叠空间。作为李群 Spin(n) 及其李代数和特殊正交群 SO(n) 有相同的维数 n(n − 1)/2。

Spin(n) 可以构造为克利福德代数 Cℓ(n) 可逆元群的一个子群。Spin(n) 由所有写成个偶数个单位向量的克利福德乘积的元素生成。对应到 SO(n) 中恰是沿着垂直于这偶数个向量的超平面反射的复合。

## 巧合同构

Spin(1) = O(1) = Z2
Spin(2) = U(1) = SO(2) = S1
Spin(3) = Sp(1) = SU(2) = HU(1) = S3
Spin(4) = Sp(1) × Sp(1)
Spin(5) = Sp(2) = HU(2)
Spin(6) = SU(4)

n = 7，8 仍然有退化的同构，细节可参见 Spin(8)；对更高的维数，这样的同构完全消失。

## 不定符号

Spin(1,1) = GL(1,R)
Spin(2,1) = SL(2,R)
Spin(3,1) = SL(2,C)
Spin(2,2) = SL(2,R) × SL(2,R)
Spin(4,1) = Sp(1,1)
Spin(3,2) = Sp(4,R)
Spin(5,1) = SL(2,H)
Spin(4,2) = SU(2,2)
Spin(3,3) = SL(4,R)

## 拓扑

${\displaystyle \pi _{1}(G)\subset Z(G'),}$

${\displaystyle ({\mbox{Spin}}(p)\times {\mbox{Spin}}(q))/\{(1,1),(-1,-1)\}}$

${\displaystyle \pi _{1}({\mbox{Spin}}(p,q))={\begin{cases}\{0\}&(p,q)=(1,1){\mbox{ or }}(1,0)\\\{0\}&p>2,q=0,1\\\mathbb {Z} &(p,q)=(2,0){\mbox{ or }}(2,1)\\\mathbb {Z} \times \mathbb {Z} &(p,q)=(2,2)\\\mathbb {Z} &p>2,q=2\\\mathbb {Z} _{2}&p>2,q>2\\\end{cases}}}$

${\displaystyle p,q>2}$，这意味着映射 ${\displaystyle \pi _{1}({\mbox{Spin}}(p,q))\to \pi _{1}(SO(p,q))}$${\displaystyle 1\in \mathbb {Z} _{2}}$ 映到 ${\displaystyle (1,1)\in \mathbb {Z} _{2}\times \mathbb {Z} _{2}}$ 给出； 对 p=2，q>2，映射由 ${\displaystyle 1\in \mathbb {Z} \to (1,1)\in \mathbb {Z} \times \mathbb {Z} _{2}}$ ；最后，对 p = q = 2， ${\displaystyle (1,0)\in \mathbb {Z} \times \mathbb {Z} }$ 映到 ${\displaystyle (1,1)\in \mathbb {Z} \times \mathbb {Z} }$${\displaystyle (0,1)}$ 映到 ${\displaystyle (1,-1)}$

## 參考文獻

• F.Reece Harvey, Spinors and Calibrations, Academic Press, Inc., 1990.
• Pertti Lounesto, Clifford Algebras and Spinors, LMSLNS 239, Cambridge University Press,1997.
• PlanetMath, Spin Groups.