In mathematics, the spin representations are particular projective representations of the orthogonal or special orthogonal groups in arbitrary dimension and signature (i.e., including indefinite orthogonal groups). More precisely, they are two equivalent representations of the spin groups, which are double coverss of the special orthogonal groups. They are usually studied over the real or complex numbers, but they can be defined over other fieldss.

Elements of a spin representation are called spinors. They play an important role in the physical description of fermions such as the electron.

The spin representations may be constructed in several ways, but typically the construction involves (perhaps only implicitly) the choice of a maximal isotropic subspace in the vector representation of the group. Over the real numbers, this usually requires using a complexification of the vector representation. For this reason, it is convenient to define the spin representations over the complex numbers first, and derive real representations by introducing real structures.

The properties of the spin representations depend, in a subtle way, on the dimension and signature of the orthogonal group. In particular, spin representations often admit invariant bilinear forms, which can be used to embed the spin groups into classical Lie groups. In low dimensions, these embeddings are surjective and determine special isomorphisms between the spin groups and more familiar Lie groups; this elucidates the properties of spinors in these dimensions.

Set-up

be a finite-dimensional vector space, equipped with a nondegenerate quadratic form. The orthogonal group and special orthogonal group are

denotes the determinant-one subgroup, rather than necessarily the identity component.

and

is connected.

, Sylvester's law of inertia and

.

The spin group is defined naturally through the Clifford algebra. With the convention used here,

be its even subalgebra denote reversion, the anti-automorphism determined by

The spin group may then be defined by

Conjugation in the Clifford algebra gives a homomorphism

In the complex case this gives an exact sequence

, the corresponding exact sequence is

. in indefinite signature. The analogous construction for the full orthogonal group uses a pin group.

identifies the Lie algebra of the spin group with the orthogonal Lie algebra:

is real, its complexification, and

.

A complex spinor module is an irreducible module:

• .

  decomposes as

• . They are called the positive and negative half-spin is called the Dirac spin representation.

. Consequently, a spin representation does not descend to a linear representation of the corresponding special orthogonal group, but its projectivization does descend to a projective representation of the spin group does not factor through the special orthogonal group.

. The existence of invariant real. It is therefore convenient first to construct the complex spin representations and then to analyze their restrictions and possible real structures.

Complex spin representations

Let with the standard quadratic form so that

The symmetric bilinear form on associated to by polarization is denoted .

Isotropic subspaces and root systems

is isotropic and generates a one-dimensional totally isotropic subspace.

A standard construction of the spin representations of begins with a choice of a pair of maximal totally isotropic subspacess (with respect to ) of with . Let us make such a choice. If or , then and both have dimension . If , then , whereas if , then , where is the 1-dimensional orthogonal complement to . The bilinear form associated to induces a pairing between and , which must be nondegenerate, because and are totally isotropic subspaces and is nondegenerate. Hence and are dual vector spaces.

More concretely, let be a basis for . Then there is a unique basis of such that

If is an matrix, then induces an endomorphism of with respect to this basis and the transpose induces a transformation of with

for all in and in . It follows that the endomorphism of , equal to on , on and zero on (if is odd), is skew,

for all in , and hence (see classical group) an element of .

Using the diagonal matrices in this construction defines a Cartan subalgebra of : the rank of is , and the diagonal matrices determine an -dimensional abelian subalgebra.

Let be the basis of such that, for a diagonal matrix is the th diagonal entry of . Clearly this is a basis for . Since the bilinear form identifies , explicitly,

it is now easy to construct the root system associated to . The root spaces (simultaneous eigenspaces for the action of ) are spanned by the following elements:

1. with root

2. (which is in if

and, if is odd, and is a nonzero element of ,

Thus, with respect to the basis , the roots are the vectors in that are permutations of

together with the permutations of

if is odd.

A system of positive roots is given by and (for odd) . The corresponding simple rootss are

The positive roots are nonnegative integer linear combinations of the simple roots.

Spin representations and their weights

One construction of the spin representations of uses the exterior algebra(s)

There is an action of on such that for any element in and any in the action is given by:

where the second term is a contraction (interior multiplication) defined using the bilinear form, which pairs and . This action respects the Clifford relations , and so induces a homomorphism from the Clifford algebra of to . A similar action can be defined on , so that both and are Clifford modules.

The Lie algebra is isomorphic to the complexified Lie algebra in via the mapping induced by the covering

It follows that both and are representations of . They are actually equivalent representations, so we focus on S.

The explicit description shows that the elements of the Cartan subalgebra act on by

A basis for is given by elements of the form

for and . These clearly span weight spaces for the action of : has eigenvalue −1/2 on the given basis vector if for some , and has eigenvalue otherwise.

It follows that the weights of are all possible combinations of

and each weight space is one-dimensional. Elements of are called Dirac spinors.

When is even, is not an irreducible representation are invariant subspaces. The weights divide into those with an even number of minus signs, and those with an odd number of minus signs. Both S+ and S are irreducible representations of dimension 2m1 whose elements are called Weyl spinors. They are also known as chiral spin representations or half-spin representations. With respect to the positive root system above, the highest weights of S+ and S are

respectively. The Clifford action identifies ClnC with End(S) and the even subalgebra is identified with the endomorphisms preserving S+ and S. The other Clifford module S is isomorphic to S in this case.

When n is odd, S is an irreducible representation of so(n,C) of dimension 2m: the Clifford action of a unit vector u ∈ U is given by

and so elements of so(n,C) of the form u∧w or u∧w∗ do not preserve the even and odd parts of the exterior algebra of W. The highest weight of S is

The Clifford action is not faithful on S: ClnC can be identified with End(S) ⊕ End(S), where u acts with the opposite sign on S. More precisely, the two representations are related by the parity involution of ClnC (also known as the principal automorphism), which is the identity on the even subalgebra, and minus the identity on the odd part of ClnC. In other words, there is a linear isomorphism from S to S, which identifies the action of A in ClnC on S with the action of (A) on S.

Bilinear forms

if is a weight of S, so is . It follows that S is isomorphic to the dual representation S∗.

When n = 2m + 1 is odd, the isomorphism B: S → S∗ is unique up to scale by Schur's lemma, since S is irreducible, and it defines a nondegenerate invariant bilinear form on S via

Here invariance means that

for all in so(n,C) and , in S in other words the action of is skew with respect to . In fact, more is true: S∗ is a representation of the opposite Clifford algebra, and therefore, since ClnC only has two nontrivial simple modules S and S, related by the parity involution , there is an antiautomorphism of ClnC such that

for any A in ClnC. In fact is reversion (the antiautomorphism induced by the identity on V) for m even, and conjugation (the antiautomorphism induced by minus the identity on V) for m odd. These two antiautomorphisms are related by parity involution , which is the automorphism induced by minus the identity on V. Both satisfy () = for in so(n,C).

When n = 2m, the situation depends more sensitively upon the parity of m. For m even, a weight has an even number of minus signs if and only if does; it follows that there are separate isomorphisms B±: S± → S±∗ of each half-spin representation with its dual, each determined uniquely up to scale. These may be combined into an isomorphism B: S → S∗. For m odd, is a weight of S+ if and only if is a weight of S; thus there is an isomorphism from S+ to S∗, again unique up to scale, and its transpose provides an isomorphism from S to S+∗. These may again be combined into an isomorphism B: S → S∗.

For both m even and m odd, the freedom in the choice of B may be restricted to an overall scale by insisting that the bilinear form corresponding to B satisfies (1), where is a fixed antiautomorphism (either reversion or conjugation).

Symmetry and the tensor square

The symmetry properties of : S ⊗ S → C can be determined using Clifford algebras or representation theory. In fact much more can be said: the tensor square S ⊗ S must decompose into a direct sum of k-forms on V for various k, because its weights are all elements in h∗ whose components belong to {1,0,1}. Now equivariant linear maps S ⊗ S → ∧kV∗ correspond bijectively to invariant maps ∧kV ⊗ S ⊗ S → C and nonzero such maps can be constructed via the inclusion of ∧kV into the Clifford algebra. Furthermore, if (,) = (,) and has sign k on ∧kV then

for A in ∧kV.

If n = 2m+1 is odd then it follows from Schur's Lemma that

(both sides have dimension 22m and the representations on the right are inequivalent). Because the symmetries are governed by an involution that is either conjugation or reversion, the symmetry of the ∧2jV∗ component alternates with j. Elementary combinatorics gives

and the sign determines which representations occur in S2S and which occur in ∧2S. In particular

1. and

for v ∈ V (which is isomorphic to ∧2mV), confirming that is reversion for m even, and conjugation for m odd.

If n = 2m is even, then the analysis is more involved, but the result is a more refined decomposition: S2S±, 2S± and S+ ⊗ S can each be decomposed as a direct sum of k-forms (where for k = m there is a further decomposition into selfdual and antiselfdual m-forms).

The main outcome is a realisation of so(n,C) as a subalgebra of a classical Lie algebra on S, depending upon n modulo 8, according to the following table:For n ≤ 6, these embeddings are isomorphisms (onto sl rather than gl for n = 6):

Real representations

The complex spin representations of so(n,C) yield real representations S of so(p,q) by restricting the action to the real subalgebras. However, there are additional "reality" structures that are invariant under the action of the real Lie algebras. These come in three types.

1. There is an invariant complex antilinear map r: S → S with r2 = idS. The fixed point set of r is then a real vector subspace SR of S with SR ⊗ C = S. This is called a real structure.

2. There is an invariant complex antilinear map j: S → S with j2 = idS. It follows that the triple i, j and k:=ij make S into a quaternionic vector space SH. This is called a quaternionic structure.

3. There is an invariant complex antilinear map b: S → S∗ that is invertible. This defines a pseudohermitian bilinear form on S and is called a hermitian structure.

The type of structure invariant under so(p,q) depends only on the signature p q modulo 8, and is given by the following table.Here R, C and H denote real, hermitian and quaternionic structures respectively, and R + R and H + H indicate that the half-spin representations both admit real or quaternionic structures respectively.

Description and tables

To complete the description of real representation, we must describe how these structures interact with the invariant bilinear forms. Since n = p + q ≅ p − q mod 2, there are two cases: the dimension and signature are both even, and the dimension and signature are both odd.

The odd case is simpler, there is only one complex spin representation S, and hermitian structures do not occur. Apart from the trivial case n = 1, S is always even-dimensional, say dim S = 2N. The real forms of so(2N,C) are so(K,L) with K + L = 2N and so∗(N,H), while the real forms of sp(2N,C) are sp(2N,R) and sp(K,L) with K + L = N. The presence of a Clifford action of V on S forces K = L in both cases unless pq = 0, in which case KL=0, which is denoted simply so(2N) or sp(N). Hence the odd spin representations may be summarized in the following table.(†) is even for and for , this is .

The even-dimensional case is similar. For , the complex half-spin representations are even-dimensional. We have additionally to deal with hermitian structures and the real forms of , which are , with , and . The resulting even spin representations are summarized as follows.(*) For , we have instead

(†) is even for and for (which includes with ), we have instead

The low-dimensional isomorphisms in the complex case have the following real forms.

Notes

References

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