Spin group

In mathematics the spin group Spin("n") is the double cover of the special orthogonal group , such that there exists a short exact sequence of Lie groups (when )
As a Lie group, Spin("n") therefore shares its dimension, , and its Lie algebra with the special orthogonal group.
For , 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 −.
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 zero-dimensional space; but of course, space is not zero-dimensional, 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". The Clifford algebra is the quotient of the tensor algebra T"V" of "V" by a two-sided ideal. The tensor algebra (over the reals) may be written as 
The Clifford algebra Cl("V") is then the quotient algebra
where formula_4 is the quadratic form applied to a vector formula_5. The resulting space is naturally graded (as a vector space), and can be written as 
where formula_7 and formula_8. The spin algebra formula_9 is defined as
where the last is a short-hand 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 formula_11 of the special orthogonal group.
The pin group formula_12 is a subgroup of formula_13's Clifford group of all elements of the form 
where each formula_15 is of unit length: formula_16
The spin group is then defined as 
where 
formula_18
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 two-component (Weyl) spinors, constructed below.
If the set formula_19 are an orthonormal basis of the (real) vector space "V", then the quotient above endows the space with a natural anti-commuting structure:
which follows by considering formula_22 for formula_23. This anti-commutation 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 anti-commuting as a by-product of the Clifford algebra construction. This anti-commutation 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.
For a quadratic space "V", a double covering of SO("V") by Spin("V") can be given explicitly, as follows. Let formula_19 be an orthonormal basis for "V". Define an antiautomorphism formula_25 by
This can be extended to all elements of formula_27 by linearity. It is an antihomomorphism since
Observe that Pin("V") can then be defined as all elements formula_29 for which
Now define the automorphism formula_31 which on degree 1 elements is given by
and let formula_33 denote formula_34, which is an antiautomorphism of Cl("V"). With this notation, an explicit double covering is the homomorphism formula_35 given by
where formula_37. When "a" has degree 1 (i.e. formula_38), formula_39 corresponds a reflection across the hyperplane orthogonal to "a"; this follows from the anti-commuting property of the Clifford algebra.
This gives a double covering of both O("V") by Pin("V") and of SO("V") by Spin("V") because formula_40 gives the same transformation as formula_41.
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 an even number, its complexification is formula_42. It can be written as the direct sum of a subspace formula_43 of spinors and a subspace formula_44 of anti-spinors:
The space formula_43 is spanned by the spinors
formula_47
for formula_48 and the complex conjugate spinors span formula_44. It is straightforward to see that the spinors anti-commute, and that the product of a spinor and anti-spinor is a scalar.
The spinor space is defined as the exterior algebra formula_50. The (complexified) Clifford algebra acts naturally on this space; the (complexified) spin group corresponds to the length-preserving endomorphisms. There is a natural grading on the exterior algebra: the product of an odd number of copies of formula_43 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.
Complex case.
The SpinC group is defined by the exact sequence 
It is a multiplicative subgroup of the complexification formula_53 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 formula_55 identifies with .
This has important applications in 4-manifold theory and Seiberg–Witten theory. In physics, the Spin group is appropriate for describing uncharged fermions, while the SpinC group is used to describe electrically charged fermions. In this case, the U(1) symmetry is specifically the gauge group of electromagnetism.
Exceptional isomorphisms.
In low dimensions, there are isomorphisms among the classical Lie groups called "exceptional 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. Writing R for the reals, C for the complex numbers, H for the quaternions and the general understanding that Cl("n") is a short-hand for Cl(R"n") and that Spin("n") is a short-hand for Spin(R"n") and so on, one then has that
There are certain vestiges of these isomorphisms left over for (see Spin(8) for more details). For higher "n", these isomorphisms disappear entirely.
Indefinite signature.
In indefinite signature, the spin group is constructed through Clifford algebras in a similar way to standard spin groups. It is a double cover of , the connected component of the identity of the indefinite orthogonal group . For , is connected; for there are two connected components. As in definite signature, there are some accidental isomorphisms in low dimensions:
Note that .
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 formula_61 of "G" determine "G" entirely (note that it is not the case that formula_61 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 of SO("n").
In indefinite signature, Spin("p", "q") is not necessarily 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
This allows us to calculate the fundamental groups of Spin("p", "q"), taking "p" ≥ "q":
Thus once 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 , this implies that the map is given by going to . For , this map is given by . And finally, for , is sent to and is sent to .
Center.
The center of the spin groups, for , (complex and real) are given as follows:
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 ), then Spin is the "maximal" group in the sequence, and one has a sequence of three groups,
splitting by parity yields:
which are the three compact real forms (or two, if ) of the compact Lie algebra formula_65
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 are equal, but π0 and π1 may differ.
For , Spin("n") is simply connected ( 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 and the component group of .
Whitehead tower.
The spin group appears in a Whitehead 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 3 homotopy group in Spin("n"), one obtains the infinite-dimensional 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 , 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 2"G", 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 formula_67 (since {±1} is central). As an example of these latter, given a cyclic group of odd order formula_68 in SO("n"), its preimage is a cyclic group of twice the order, formula_69 and the subgroup maps isomorphically to .
Of particular note are two series:
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.