|
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 : 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 dim=3 :Spin(4) = SU(2) × SU(2), corresponding to dim=6 :Spin(5) = Sp(2), corresponding to dim=10 :Spin(6) = SU(4), corresponding to 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）』 ■ウィキペディアで「spin group」の詳細全文を読む スポンサード リンク
|