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Pin(5)

Contents

Context

Group Theory

Spin 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 ±(5)Sp(2)ωSp(2)AAAω 2=±e Pin^\pm(5) \;\simeq\; Sp(2) \sqcup \omega Sp(2) \phantom{AAA} \omega^2 = \pm e

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

(e.g. Varlamov 99, Theorem 5)

rotation groups in low dimensions:

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.