Contents

group theory

spin geometry

string geometry

# Contents

## 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)

sp. orth. groupspin grouppin groupsemi-spin group
SO(2)Spin(2)Pin(2)
SO(3)Spin(3)
SO(4)Spin(4)
SO(5)Spin(5)Pin(5)
Spin(6)
Spin(7)
SO(8)Spin(8)SO(8)
SO(9)Spin(9)
$\vdots$$\vdots$
SO(16)Spin(16)SemiSpin(16)
SO(32)Spin(32)SemiSpin(32)

see also

## 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)

Created on May 14, 2019 at 00:14:05. See the history of this page for a list of all contributions to it.