Spin group

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 Cℓ(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

z

ℝ 2

by double phase rotation

z \mapsto u 2 z

. 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 $n$ = 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), the connected component of the identity of the indefinite 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′ the universal cover of G, there is an inclusion

\pi_1 (G) \subset \operatorname{Z}(G'),

with Z(G′) the center 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 π₁(G) determine G entirely; for instance SL(2, R) and PSL(2, R) have the same Lie algebra and same fundamental group 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, Spin(p, q) is not connected, and in general the identity component, Spin₀(p, q), is not simply connected, thus it is not a universal cover. The fundamental group is most easily understood by considering the maximal compact subgroup of SO(p, q) , which is SO(p) × SO(q), and noting that rather than being the product of the 2-fold covers (hence a 4-fold cover), Spin(p, q) is the "diagonal" 2-fold cover – it is a 2-fold quotient of the 4-fold cover. Explicitly, the maximal compact connected subgroup of Spin(p, q) is

Spin(p) × Spin(q)/{(1, 1), (−1, −1)}.

This allows us to calculate the fundamental groups of Spin(p, q), taking p ≥ q:

\pi_1(\mbox{Spin}(p,q)) = \begin{cases} \{0\} & (p,q)=(1,1) \mbox{ or } (1,0) \\ \{0\} & p > 2, q = 0,1 \\ \mathbf{Z} & (p,q)=(2,0) \mbox{ or } (2,1) \\ \mathbf{Z} \times \mathbf{Z} & (p,q) = (2,2) \\ \mathbf{Z} & p > 2, q=2 \\ \mathrm{Z}_2 & p, q >2\\ \end{cases}

Thus once p, q > 2 the fundamental group is Z₂, as it is a 2-fold quotient of a product of two universal covers.

The maps on fundamental groups are given as follows. For p, q > 2, this implies that the map π₁(Spin(p, q)) → π₁(SO(p, q)) is given by 1 ∈ Z₂ going to (1,1) ∈ Z₂ × Z₂. For p = 2, q > 2 , this map is given by 1 ∈ Z → (1,1) ∈ Z × Z₂. And finally, for p = q = 2 , (1,0) ∈ Z × Z is sent to (1,1) ∈ Z × Z and (0, 1) is sent to (1, −1).

Center

The center of the spin groups, for n≥3, (complex and real) are given as follows:^[3]

\begin{align} \operatorname{Z}(\operatorname{Spin}(n,\mathbf{C})) &= \begin{cases} \mathrm{Z}_2 & n = 2k+1\\ \mathrm{Z}_4 & n = 4k+2\\ \mathrm{Z}_2 \oplus \mathrm{Z}_2 & n = 4k\\ \end{cases} \\ \operatorname{Z}(\operatorname{Spin}(p,q)) &= \begin{cases} \mathrm{Z}_2 & n = 2k+1,\\ \mathrm{Z}_2 & n = 2k, \text{ and } p, q \text{ odd}\\ \mathrm{Z}_4 & n = 2k, \text{ and } p, q \text{ even}\\ \end{cases} \end{align}

Quotient groups

Quotient groups can be obtained from a spin group by quotienting out by a subgroup of the center, with the spin group then being a covering group of the resulting quotient, and both groups having the same Lie algebra.

Quotienting out by the entire center yields the minimal such group, the projective special orthogonal group, which is centerless, while quotienting out by {±1} yields the special orthogonal group – if the center equals {±1} (namely in odd dimension), these two quotient groups agree. If the spin group is simply connected (as Spin(n) is for n > 2), then Spin is the maximal group in the sequence, and one has a sequence of three groups,

Spin(n) → SO(n) → PSO(n),

splitting by parity yields:

Spin(2n) → SO(2n) → PSO(2n),

Spin(2n+1) → SO(2n+1) = PSO(2n+1),

which are the three compact real forms (or two, if SO = PSO ) of the compact Lie algebra $\mathfrak{so} (n, \mathbf{R}).$

The homotopy groups of the cover and the quotient are related by the long exact sequence of a fibration, with discrete fiber (the fiber being the kernel) – thus all homotopy groups for k > 1 are equal, but π₀ and π₁ may differ.

For n > 2, Spin(n) is simply connected (π₀ = π₁ = {1} is trivial), so SO(n) is connected and has fundamental group Z₂ while PSO(n) is connected and has fundamental group equal to the center of Spin(n).

In indefinite signature the covers and homotopy groups are more complicated – Spin(p, q) is not simply connected, and quotienting also affects connected components. The analysis is simpler if one considers the maximal (connected) compact SO(p) × SO(q) ⊂ SO(p, q) and the component group of Spin(p, q).

Discrete subgroups

Discrete subgroups of the spin group can be understood by relating them to discrete subgroups of the special orthogonal group (rotational point groups).

Given the double cover Spin(n) → SO(n), by the lattice theorem, there is a Galois connection between subgroups of Spin(n) and subgroups of SO(n) (rotational point groups): the image of a subgroup of Spin(n) is a rotational point group, and the preimage of a point group is a subgroup of Spin(n), and the closure operator on subgroups of Spin(n) is multiplication by {±1}. These may be called "binary point groups"; most familiar is the 3-dimensional case, known as binary polyhedral groups.

Concretely, every binary point group is either the preimage of a point group (hence denoted 2G, for the point group G), or is an index 2 subgroup of the preimage of a point group which maps (isomorphically) onto the point group; in the latter case the full binary group is abstractly $\mathrm{C}_2 \times G$ (since {±1} is central). As an example of these latter, given a cyclic group of odd order $\mathrm{C}_{2k+1}$ in SO(n), its preimage is a cyclic group of twice the order, $\mathrm{C}_{4k+2} \cong \mathrm{C}_{2k+1} \times \mathrm{C}_2,$ and the subgroup C_2k+1 < Spin(n) maps isomorphically to C_2k+1 < SO(n).

Of particular note are two series:

higher binary tetrahedral groups, corresponding to the 2-fold cover of symmetries of the n-simplex.
This group can also be considered as the double cover of the symmetric group, 2⋅ A_n → A_n, with the alternating group being the (rotational) symmetry group of the n-simplex.
higher binary octahedral groups, corresponding to the 2-fold covers of the hyperoctahedral group (symmetries of the hypercube, or equivalently of its dual, the cross-polytope).

For point groups that reverse orientation, the situation is more complicated, as there are two pin groups, so there are two possible binary groups corresponding to a given point group.

Complex case

Main article: Spin_structure § Spinc_structures

The spin^c group is defined by the exact sequence

1 \to \mathrm{Z}_2 \to {\mathrm {Spin}}^{\mathbb C}(n) \to {\mathrm {SO}}(n)\times {\mathrm U}(1) \to 1.

This has important applications in 4-manifold theory and Seiberg–Witten theory.

References

↑ Lawson, H. Blaine; Michelsohn, Marie-Louise (1989). Spin Geometry. Princeton University Press. ISBN 978-0-691-08542-5. page 14
↑ Friedrich, Thomas (2000), Dirac Operators in Riemannian Geometry, American Mathematical Society, ISBN 978-0-8218-2055-1 page 15
↑ (Varadarajan 2004, p. 208)