nLab Pin(5)

Idea

The pin group in dimension 5.

Properties

Exceptional isomorphisms

Proposition

The exceptional isomorphism Spin(5) $\simeq$ Sp(2) (this Prop.) generalizes to

$$Pin^\pm(5) \;\simeq\; Sp(2) \sqcup \omega Sp(2) \phantom{AAA} \omega^2 = \pm e$$

where $\omega \in Z\big( Pin^+(5)\big)$ is an element in the center which, for $Pin^+(5)$, squares to the the neutral element $e$ (corresponding to the Clifford algebra element $+1$) or, for $Pin^-(5)$, to $-e$ (the Clifford algebra element $-1$).

(e.g. Varlamov 99, Theorem 5)

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

• Vadim V. Varlamov, Fundamental Automorphisms of Clifford Algebras and an Extension of Dabrowski Pin Groups, Hadronic J. 22 (1999) 497-533 (arXiv:math-ph/9904038v2)