nLab 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:

Dynkin labelsp. orth. groupspin grouppin groupsemi-spin group
SO(2)Spin(2)Pin(2)
B1SO(3)Spin(3)Pin(3)
D2SO(4)Spin(4)Pin(4)
B2SO(5)Spin(5)Pin(5)
D3SO(6)Spin(6)
B3SO(7)Spin(7)
D4SO(8)Spin(8)SO(8)
B4SO(9)Spin(9)
D5SO(10)Spin(10)
B5SO(11)Spin(11)
D6SO(12)Spin(12)
\vdots\vdots
D8SO(16)Spin(16)SemiSpin(16)
\vdots\vdots
D16SO(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 04:14:05. See the history of this page for a list of all contributions to it.