- Spin group
In
mathematics the spin group Spin("n") is the double cover of thespecial orthogonal group SO("n"), such that there exists ashort exact sequence ofLie group s:1 o mathbb{Z}_2 o operatorname{Spin}(n) o operatorname{SO}(n) o 1.For "n" > 2, Spin("n") issimply connected and so coincides with theuniversal cover of SO("n"). As a Lie group Spin("n") therefore shares its dimension, "n"("n" − 1)/2, and itsLie algebra with the special orthogonal group.Spin("n") can be constructed as a
subgroup of the invertible elements in theClifford algebra "C"ℓ("n").Accidental isomorphisms
In low dimensions, there are
isomorphism s among the classical Lie groups called "accidental isomorphisms". For instance, there are isomorphisms between low dimensional spin groups and certain classical Lie groups, due to low dimensional isomorphisms between theRoot systems of the different families of simple Lie algebras. Specifically, we have:Spin(1) = O(1):Spin(2) = U(1) = SO(2):Spin(3) = Sp(1) = SU(2):Spin(4) = Sp(1) × Sp(1):Spin(5) = Sp(2):Spin(6) = SU(4)
There are certain vestiges of these isomorphisms left over for "n" = 7,8 (see
Spin(8) for more details). For higher "n", these isomorphisms disappear entirely.Indefinite signature
In indefinite signature, the spin group Spin("p","q") is constructed through
Clifford algebras in a similar way to standard spin groups. It is a connected double cover of SO0("p","q"), theconnected component of the identity of theindefinite orthogonal group SO("p","q") (there are a variety of conventions on the connectedness of Spin("p","q"); in this article, it is taken to be connected for "p"+"q">2). As in definite signature, there are some accidental isomorphisms in low dimensions::Spin(1,1) = GL(1,R)
:Spin(2,1) = SL(2,R)
:Spin(3,1) = SL(2,C):Spin(2,2) = SL(2,R) × SL(2,R)
:Spin(4,1) = Sp(1,1):Spin(3,2) = Sp(4,R)
:Spin(5,1) = SL(2,H):Spin(4,2) = SU(2,2):Spin(3,3) = SL(4,R)
Note that Spin("p","q") = Spin("q","p").
Topological considerations
Connected and
simply connected Lie groups are classified by their Lie algebra. So if "G" is a connected Lie group with a simple Lie algebra, with "G"′ theuniversal cover of "G", there is an inclusion:pi_1 (G) subset Z(G'),
with "Z"("G"′) the centre of "G"′. This inclusion and the Lie algebra mathfrak{g} of "G" determine "G" entirely (note that it is not the fact that mathfrak{g} and pi_1 (G) determine "G" entirely; for instance SL(2,R) and PSL(2,R) have the same Lie algebra and same fundamental group mathbb{Z}, but are not isomorphic).
The definite signature Spin("n") are all
simply connected for ("n">2), so they are the universal coverings for SO("n"). In indefinite signature, the maximal compact connected subgroup of Spin("p","q") is:mbox{Spin}(p) imes mbox{Spin}(q))/ {(1,1),(-1,-1)}.
This allows us to calculate the
fundamental groups of Spin("p","q")::pi_1(mbox{Spin}(p,q)) = egin{cases}{0} & (p,q)=(1,1) mbox{ or } (1,0) \{0} & p > 2, q = 0,1 \mathbb{Z} & (p,q)=(2,0) mbox{ or } (2,1) \mathbb{Z} imes mathbb{Z} & (p,q) = (2,2) \mathbb{Z} & p > 2, q=2 \mathbb{Z}_2 & p >2, q >2 \end{cases}
For p,q>2, this implies that the map pi_1(mbox{Spin}(p,q)) o pi_1(SO(p,q)) is given by 1 in mathbb{Z}_2 going to 1,1) in mathbb{Z}_2 imes mathbb{Z}_2. For "p"=2, "q">2, this map is given by 1 in mathbb{Z} o (1,1) in mathbb{Z} imes mathbb{Z}_2. And finally, for "p"="q"=2, 1,0) in mathbb{Z} imes mathbb{Z} is sent to 1,1) in mathbb{Z} imes mathbb{Z} and 0,1) is sent to 1,-1).
ee also
*
Pin group
*Spinor
*Spinor bundle
*Anyon
*Spin structure
*Clifford algebra
*Orientation entanglement
*Complex Spin Group
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