Spin group
In mathematics the spin group Spin(n)[1][2] is the double cover of the special orthogonal group SO(n) = SO(n, R), such that there exists a short exact sequence of Lie groups (when n ≠ 2)
Algebraic structure → Group theory Group theory 



Infinite dimensional Lie group

As a Lie group, Spin(n) therefore shares its dimension, n(n − 1)/2, 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 nontrivial 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). A distinct article discusses the spin representations.
Motivation and physical interpretation
The spin group is used in physics to describe the symmetries of (electrically neutral, uncharged) fermions. Its complexification, Spinc, is used to describe electrically charged fermions, most notably the electron. Strictly speaking, the spin group describes a fermion in a zerodimensional space; but of course, space is not zerodimensional, and so the spin group is used to define spin structures on (pseudo)Riemannian manifolds: the spin group is the structure group of a spinor bundle. The affine connection on a spinor bundle is the spin connection; the spin connection is useful as it can simplify and bring elegance to many intricate calculations in general relativity. The spin connection in turn enables the Dirac equation to be written in curved spacetime (effectively in the tetrad coordinates), which in turn provides a footing for quantum gravity, as well as a formalization of Hawking radiation (where one of a pair of entangled, virtual fermions fall past the event horizon, and the other does not). In short, the spin group is a vital cornerstone, centrally important for understanding advanced concepts in modern theoretical physics. In mathematics, the spin group is interesting in its own right: not only for these reasons, but for many more.
Construction
Construction of the Spin group often starts with the construction of a Clifford algebra over a real vector space V with a definite quadratic form q.[3] The Clifford algebra is the quotient of the tensor algebra TV of V by a twosided ideal. The tensor algebra (over the reals) may be written as
The Clifford algebra Cl(V) is then the quotient algebra
where is the quadratic form applied to a vector . The resulting space is naturally graded, and can be written as
where and . The spin algebra is defined as
where the last is a shorthand for V being a real vector space of real dimension n. It is a Lie algebra; it has a natural action on V, and in this way can be shown to be isomorphic to the Lie algebra of the special orthogonal group.
The pin group is a subgroup of 's Clifford group of all elements of the form
where each is of unit length:
The spin group is then defined as
where is the subspace generated by elements that are the product of an even number of vectors. That is, Spin(V) consists of all elements of Pin(V), given above, with the restriction to k being an even number. The restriction to the even subspace is key to the formation of twocomponent (Weyl) spinors, constructed below.
If the set are an orthonormal basis of the (real) vector space V, then the quotient above endows the space with a natural anticommuting structure:
 for
which follows by considering for . This anticommutation turns out to be of importance in physics, as it captures the spirit of the Pauli exclusion principle for fermions. A precise formulation is out of scope, here, but it involves the creation of a spinor bundle on Minkowski spacetime; the resulting spinor fields can be seen to be anticommuting as a byproduct of the Clifford algebra construction. This anticommutation property is also key to the formulation of supersymmetry. The Clifford algebra and the spin group have many interesting and curious properties, some of which are listed below.
Double covering
A double covering of SO(n) by Spin(n) can be given explicitly, as follows. Let be an orthonormal basis for V. Define an antiautomorphism by
This can be extended to all elements of by homomorphism:
Observe that Spin(V) can then be defined as all elements for which
With this notation, an explicit double covering is the homomorphism given by
where . The above gives a double covering of both O(n) by Pin(n) and of SO(n) by Spin(n) because gives the same transformation as . With a small amount of work, it can be seen that corresponds to reflection across a hyperplane; this follows from the anticommuting property of the Clifford algebra.
Spinor space
It is worth reviewing how spinor space and Weyl spinors are constructed, given this formalism. Given a real vector space V of dimension n = 2m an even number, its complexification is . It can be written as the direct sum of a subspace of spinors and a subspace of antispinors:
The space is spanned by the spinors for and the complex conjugate spinors span . It is straightforward to see that the spinors anticommute, and that the product of a spinor and antispinor is a scalar.
The spinor space is defined as the exterior algebra . The (complexified) Clifford algebra acts naturally on this space; the (complexified) spin group corresponds to the lengthpreserving endomorphisms. There is a natural grading on the exterior algebra: the product of an odd number of copies of correspond to the physics notion of fermions; the even subspace corresponds to the bosons. The representations of the action of the spin group on the spinor space can be built in a relatively straightforward fashion.[3]
Complex case
The Spin^{C} group is defined by the exact sequence
It is a multiplicative subgroup of the complexification of the Clifford algebra, and specifically, it is the subgroup generated by Spin(V) and the unit circle in C. Alternately, it is the quotient
where the equivalence identifies (a, u) with (−a, −u).
This has important applications in 4manifold theory and Seiberg–Witten theory. In physics, the Spin group is appropriate for describing uncharged fermions, while the Spin^{C} group is used to describe electrically charged fermions. In this case, the U(1) symmetry is specifically the gauge group of electromagnetism.
Accidental isomorphisms
In low dimensions, there are isomorphisms among the classical Lie groups called accidental isomorphisms. For instance, there are isomorphisms between lowdimensional spin groups and certain classical Lie groups, owing to lowdimensional isomorphisms between the root systems (and corresponding isomorphisms of Dynkin diagrams) of the different families of simple Lie algebras. Writing R for the reals, C for the complex numbers, H for the quaternions and the general understanding that Cl(n) is a shorthand for Cl(R^{n}) and that Spin(n) is a shorthand for Spin(R^{n}) and so on, one then has that[3]
 Cl(1) = C the complex numbers
 Pin(1) = {+i, −i, +1, −1}
 Spin(1) = O(1) = {+1, −1} the orthogonal group of dimension zero.

 Cl(2) = H the quaternions
 Spin(2) = U(1) = SO(2), which acts on z in R^{2} by double phase rotation z ↦ u^{2}z. dim = 1


 Cl(4)= M(2, H) the twobytwo matrices with quaternionic coefficients
 Spin(4) = SU(2) × SU(2), corresponding to . dim = 6

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 double cover of SO_{0}(p, q), the connected component of the identity of the indefinite orthogonal group SO(p, q). For p + q > 2, Spin(p, q) is connected; for (p, q) = (1, 1) there are two connected components.[4]^{:193} 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)
 Spin(6, 2) = SU(2, 2, H)
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
with Z(G′) the center of G′. This inclusion and the Lie algebra of G determine G entirely (note that it is not the case that and π_{1}(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 of SO(n).
In indefinite signature, Spin(p, q) is not necessarily connected, and in general the identity component, Spin_{0}(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 2fold covers (hence a 4fold cover), Spin(p, q) is the "diagonal" 2fold cover – it is a 2fold quotient of the 4fold 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:
Thus once p, q > 2 the fundamental group is Z_{2}, as it is a 2fold 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 π_{1}(Spin(p, q)) → π_{1}(SO(p, q)) is given by 1 ∈ Z_{2} going to (1, 1) ∈ Z_{2} × Z_{2}. For p = 2, q > 2, this map is given by 1 ∈ Z → (1,1) ∈ Z × Z_{2}. 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).
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
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 π_{0} and π_{1} may differ.
For n > 2, Spin(n) is simply connected (π_{0} = π_{1} = Z_{1} is trivial), so SO(n) is connected and has fundamental group Z_{2} 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).
Postnikov tower
The spin group appears in a Postnikov tower anchored by the orthogonal group:
The tower is obtained by successively removing (killing) homotopy groups of increasing order. This is done by constructing short exact sequences starting with an Eilenberg–MacLane space for the homotopy group to be removed. Killing the π_{3} homotopy group in Spin(n), one obtains the infinitedimensional string group String(n).
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 3dimensional 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 (since {±1} is central). As an example of these latter, given a cyclic group of odd order in SO(n), its preimage is a cyclic group of twice the order, and the subgroup Z_{2k+1} < Spin(n) maps isomorphically to Z_{2k+1} < SO(n).
Of particular note are two series:
 higher binary tetrahedral groups, corresponding to the 2fold cover of symmetries of the nsimplex; 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 nsimplex.
 higher binary octahedral groups, corresponding to the 2fold covers of the hyperoctahedral group (symmetries of the hypercube, or equivalently of its dual, the crosspolytope).
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.
See also
Related groups
 Pin group Pin(n) – twofold cover of orthogonal group, O(n)
 Metaplectic group Mp(2n) – twofold cover of symplectic group, Sp(2n)
References
 Lawson, H. Blaine; Michelsohn, MarieLouise (1989). Spin Geometry. Princeton University Press. ISBN 9780691085425. page 14
 Friedrich, Thomas (2000), Dirac Operators in Riemannian Geometry, American Mathematical Society, ISBN 9780821820551 page 15
 Jürgen Jost, Riemannian Geometry and Geometric Analysis, (2002) Springer Verlag ISBN 3540426272 (See Chapter 1.)
 Varadarajan, V. S. (2004). Supersymmetry for mathematicians : an introduction. Providence, R.I.: American Mathematical Society. ISBN 0821835742. OCLC 55487352.
Further reading
 Karoubi, Max (2008). KTheory. Springer. pp. 210–214. ISBN 9783540798897.