nLab Spin(4)

Idea

$Spin(4)$ is the spin group in dimension 4, the double cover of SO(4).

Properties

Exceptional isomorphisms

Let $\mathbb{H}$ be the real vector space underlying the quaternions. Notice that Spin(3) is the group of unit quaternions under quaternion multiplication

$$Spin(3) \simeq S(\mathbb{H}) \,.$$

This induces a group homomorphism

(1)$$\array{ Spin(3) \times Spin(3) &\longrightarrow& O(4) \\ (e_1, e_2) &\mapsto& \big( q \mapsto e_1 \cdot q \cdot \overline{e_2} \big) }$$

Proposition

The group homomorphism (1) is a double cover and hence exhibits an isomorphism between Spin(4) and the direct product group of Spin(3) with itself:

(2)$$\vartheta \;\colon\; Spin(3) \times Spin(3) \overset{\simeq}{\longrightarrow} Spin(4)$$

Since the action of Spin(3) on the imaginary quaternions $\mathbb{H}_{im} \simeq_{\mathbb{R}} \mathbb{R}^3$ is the conjugation action by unit quaternions, it follows in particular, that the canonical inclusion of Spin(3) into Spin(4) is given by the diagonal morphsm with respect to the identification (2):

(3)$$\array{ Spin(3) &\hookrightarrow& Spin(4) \\ & {}_{\mathllap{\Delta}}\searrow & \Big\downarrow^{ \mathrlap{\simeq} } \\ && Spin(3) \times Spin(3) }$$

(e.g. Berger 87, Thm. 8.9.8, Garrett 13, §2.3)

In summary:

Proposition

There is a commuting diagram of Lie groups of the form

$$\array{ ( q_1, q_2 ) &\mapsto& (x \mapsto q_1 \cdot x \cdot \overline{q}_2) \\ Sp(1) \times Sp(1) &\overset{\simeq}{\longrightarrow}& Spin(4) \\ \big\downarrow && \big\downarrow \\ Sp(1)\cdot Sp(1) &\overset{\simeq}{\longrightarrow}& SO(4) }$$

where

1. in the top left we have Sp(1) = Spin(3),

2. in the top right we have Spin(4),

3. in the bottom left we have Sp(1).Sp(1)

4. in the bottom right we have SO(4)

5. the horizontal morphism assigns the conjugation action of unit quaternions, as indicated,

6. the right vertical morphism is the defining double cover,

7. the left vertical morphism is the defining quotient group-projection.

Remark

(exceptional isomorphism via Dynkin diagrams)

Under the classification of simple Lie groups via Dynkin diagrams, and via the further exceptional isomorphism Spin(3) $\simeq$ SU(2), the exceptional isomorphism (2) corresponds to the coincidence of the D3 with the A3 diagrams, both with their central node removed:

Euler class and Pontryagin class

Proposition

(integral cohomology of classifying space/universal characteristic classes)

The integral cohomology ring of the classifying space of Spin(3) is freely generated from $1/4$th of the first Pontryagin class:

$$H^\bullet \big( B Spin(3), \mathbb{Z} \big) \;\simeq\; \mathbb{Z} \big[ \tfrac{1}{4}p_1 \big]$$

Moreover, the integral cohomology ring of the classifying space of Spin(4) is freely generated from the first fractional Pontryagin class $\tfrac{1}{2}p_1$ and the combination $\tfrac{1}{2}\big( \chi + \tfrac{1}{2}p_1 \big)$, where $\chi$ is the Euler class:

$$H^\bullet \big( B Spin(4), \mathbb{Z} \big) \;\simeq\; \mathbb{Z} \big[ \tfrac{1}{2}p_1 \,, \tfrac{1}{2}\big( \chi+ \tfrac{1}{2}p_1 \big) \big]$$

Finally, under the exceptional isomorphism (1) $\vartheta \;\colon\; Spin(3) \times Spin(3) \overset{\simeq}{\to} Spin(4)$ these classes are related by

$$\begin{aligned} \vartheta^\ast \left( \tfrac{1}{2}p_1 \right) & = \phantom{-} \tfrac{1}{4}p_1 \otimes 1 + 1 \otimes \tfrac{1}{4} p_1 \\ \vartheta^\ast \Big( \tfrac{1}{2} \big( \chi + \tfrac{1}{2} p_1 \big) \Big) & = \phantom{-} \phantom{ 1 \otimes \tfrac{1}{4}p_1 + } 1 \otimes \tfrac{1}{4}p_1 \\ \text{hence} \phantom{AAAA} \vartheta^\ast\big( \chi \big) \phantom{ + \tfrac{1}{2} p_1 \big) \Big) } & = - \tfrac{1}{4}p_1 \otimes 1 + 1 \otimes \tfrac{1}{4} p_1 \end{aligned}$$

Therefore, under the canonical diagonal inclusion $\iota \colon Spin(3) \overset{\Delta}{\hookrightarrow} Spin(3) \times Spin(3) \simeq Spin(4)$ (3) we have

$$\begin{aligned} \iota^\ast \left( \tfrac{1}{2}p_1 \right) & = \tfrac{1}{2}p_1 \\ \iota^\ast \big( \chi \big) & = 0 \end{aligned}$$

(e.g. Čadek-Vanžura 98, Lemma 2.1)

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Related concepts

rotation groups in low dimensions:

Dynkin label	sp. orth. group	spin group	pin group	semi-spin group
	SO(2)	Spin(2)	Pin(2)
B1	SO(3)	Spin(3)	Pin(3)
D2	SO(4)	Spin(4)	Pin(4)
B2	SO(5)	Spin(5)	Pin(5)
D3	SO(6)	Spin(6)
B3	SO(7)	Spin(7)
D4	SO(8)	Spin(8)		SO(8)
B4	SO(9)	Spin(9)
D5	SO(10)	Spin(10)
B5	SO(11)	Spin(11)
D6	SO(12)	Spin(12)
	$\vdots$	$\vdots$
D8	SO(16)	Spin(16)		SemiSpin(16)
	$\vdots$	$\vdots$
D16	SO(32)	Spin(32)		SemiSpin(32)

see also

• Spin(5).Spin(3)

• finite rotation groups

• ADE-classification

References

• Marcel Berger, Section 8.9 of: Geometry I, Springer 1987 (doi:10.1007/978-3-540-93815-6)

• Martin Čadek, Jiří Vanžura, On 4-fields and 4-distributions in 8-dimensional vector bundles over 8-complexes, Colloquium Mathematicum 1998, 76 (2), pp 213-228 (web)

• Paul Garrett, Sporadic isogenies to orthogonal groups (July 2013) [pdf, pdf ]