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spin group : ウィキペディア英語版
spin group

In mathematics the spin group Spin(''n'') 〔 page 14〕〔 page 15〕 is the double cover of the special orthogonal group , such that there exists a short exact sequence of Lie groups
:1 \to \mathrm_2 \to \operatorname(n) \to \operatorname(n) \to 1.
As a Lie group, Spin(''n'') therefore shares its dimension, , and its Lie algebra with the special orthogonal group.
For ''n'' > 2, Spin(''n'') is simply connected and so coincides with the universal cover of SO(''n'').
The non-trivial element of the kernel is denoted −1 , which should not be confused with the orthogonal transform of reflection through the origin, generally denoted −''I'' .
Spin(''n'') can be constructed as a subgroup of the invertible elements in the Clifford algebra ''C''ℓ(''n'').
==Accidental isomorphisms==
In low dimensions, there are isomorphisms among the classical Lie groups called ''accidental isomorphisms''. For instance, there are isomorphisms between low-dimensional spin groups and certain classical Lie groups, owing to low-dimensional isomorphisms between the root systems (and corresponding isomorphisms of Dynkin diagrams) of the different families of simple Lie algebras. Specifically, we have
:Spin(1) = O(1).     dim=0
:Spin(2) = U(1) = SO(2), which acts on in by double phase rotation .     dim=1
:Spin(3) = Sp(1) = SU(2), corresponding to B_1 \cong A_1.     dim=3
:Spin(4) = SU(2) × SU(2), corresponding to D_2 \cong A_1 \times A_1.     dim=6
:Spin(5) = Sp(2), corresponding to B_2 \cong C_2.     dim=10
:Spin(6) = SU(4), corresponding to D_3 \cong A_3.     dim=15
There are certain vestiges of these isomorphisms left over for  = 7, 8 (see Spin(8) for more details). For higher , these isomorphisms disappear entirely.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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