For n > 2, Spin(n) is simply connected and so coincides with the universal cover of SO(n).
Spin(n) can be constructed as a subgroup of the invertible elements in the Clifford algebra Cl(n).
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.
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 falls past the event horizon, and the other does not).
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
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.
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.
[citation needed] The Clifford algebra and the spin group have many interesting and curious properties, some of which are listed below.
The spin groups can be constructed less explicitly but without appealing to Clifford algebras.
is a set with two elements, and one can be chosen without loss of generality to be the identity.
For a quadratic space V, a double covering of SO(V) by Spin(V) can be given explicitly, as follows.
Given a real vector space V of dimension n = 2m an even number, its complexification is
There is a natural grading on the exterior algebra: the product of an odd number of copies of
The representations of the action of the spin group on the spinor space can be built in a relatively straightforward fashion.
[3] The SpinC 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
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.
It is a double cover of SO0(p, q), the connected component of the identity of the indefinite orthogonal group SO(p, q).
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).
Explicitly, the maximal compact connected subgroup of Spin(p, q) is This allows us to calculate the fundamental groups of SO(p, q), taking p ≥ q: Thus once p, q > 2 the fundamental group is Z2, as it is a 2-fold quotient of a product of two universal covers.
Then Theorem 4.41 in Hatcher tells us that there is a long exact sequence of homotopy groups
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, splitting by parity yields: 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 π0 and π1 may differ.
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).
This is done by constructing short exact sequences starting with an Eilenberg–MacLane space for the homotopy group to be removed.