Isotropic subspaces and root systems

A standard construction of the spin representations of so(n, C) begins with a choice of a pair (W, W^∗) of maximal totally isotropic subspaces (with respect to Q) of V with W ∩ W^∗ = 0. Let us make such a choice. If n = 2m or n = 2m + 1, then W and W^∗ both have dimension m. If n = 2m, then V = W ⊕ W^∗, whereas if n = 2m + 1, then V = W ⊕ U ⊕ W^∗, where U is the 1-dimensional orthogonal complement to W ⊕ W^∗. The bilinear form ⟨.,.⟩ associated to Q induces a pairing between W and W^∗, which must be nondegenerate, because W and W^∗ are totally isotropic subspaces and Q is nondegenerate. Hence W and W^∗ are dual vector spaces.

More concretely, let a₁, … a_m be a basis for W. Then there is a unique basis α₁, ... α_m of W^∗ such that

${\displaystyle \langle \alpha _{i},a_{j}\rangle =\delta _{ij}.}$

If A is an m × m matrix, then A induces an endomorphism of W with respect to this basis and the transpose A^T induces a transformation of W^∗ with

${\displaystyle \langle Aw,w^{*}\rangle =\langle w,A^{\mathrm {T} }w^{*}\rangle }$

for all w in W and w^∗ in W^∗. It follows that the endomorphism ρ_A of V, equal to A on W, −A^T on W^∗ and zero on U (if n is odd), is skew,

${\displaystyle \langle \rho _{A}u,v\rangle =-\langle u,\rho _{A}v\rangle }$

for all u, v in V, and hence (see classical group) an element of so(n, C) ⊂ End(V).

Using the diagonal matrices in this construction defines a Cartan subalgebra h of so(n, C): the rank of so(n, C) is m, and the diagonal n × n matrices determine an m-dimensional abelian subalgebra.

Let ε₁, … ε_m be the basis of h^∗ such that, for a diagonal matrix A, ε_k(ρ_A) is the kth diagonal entry of A. Clearly this is a basis for h^∗. Since the bilinear form identifies so(n, C) with ${\displaystyle \wedge ^{2}V}$, explicitly,

${\displaystyle x\wedge y\mapsto \varphi _{x\wedge y},\quad \varphi _{x\wedge y}(v)=2(\langle y,v\rangle x-\langle x,v\rangle y),\quad x\wedge y\in \wedge ^{2}V,\quad x,y,v\in V,\quad \varphi _{x\wedge y}\in {\mathfrak {so}}(n,\mathbb {C} ),}$^[1]

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

${\displaystyle a_{i}\wedge a_{j},\;i\neq j,}$ with root (simultaneous eigenvalue) ${\displaystyle \varepsilon _{i}+\varepsilon _{j}}$

${\displaystyle a_{i}\wedge \alpha _{j}}$ (which is in h if i = j) with root ${\displaystyle \varepsilon _{i}-\varepsilon _{j}}$

${\displaystyle \alpha _{i}\wedge \alpha _{j},\;i\neq j,}$ with root ${\displaystyle -\varepsilon _{i}-\varepsilon _{j},}$

and, if n is odd, and u is a nonzero element of U,

${\displaystyle a_{i}\wedge u,}$ with root ${\displaystyle \varepsilon _{i}}$

${\displaystyle \alpha _{i}\wedge u,}$ with root ${\displaystyle -\varepsilon _{i}.}$

Thus, with respect to the basis ε₁, … ε_m, the roots are the vectors in h^∗ that are permutations of

${\displaystyle (\pm 1,\pm 1,0,0,\dots ,0)}$

together with the permutations of

${\displaystyle (\pm 1,0,0,\dots ,0)}$

if n = 2m + 1 is odd.

A system of positive roots is given by ε_i + ε_j (i ≠ j), ε_i − ε_j (i < j) and (for n odd) ε_i. The corresponding simple roots are

${\displaystyle \varepsilon _{1}-\varepsilon _{2},\varepsilon _{2}-\varepsilon _{3},\ldots ,\varepsilon _{m-1}-\varepsilon _{m},\left\{{\begin{matrix}\varepsilon _{m-1}+\varepsilon _{m}&n=2m\\\varepsilon _{m}&n=2m+1.\end{matrix}}\right.}$

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

Spin representations and their weights

One construction of the spin representations of so(n, C) uses the exterior algebra(s)

${\displaystyle S=\wedge ^{\bullet }W}$ and/or ${\displaystyle S'=\wedge ^{\bullet }W^{*}.}$

There is an action of V on S such that for any element v = w + w^∗ in W ⊕ W^∗ and any ψ in S the action is given by:

${\displaystyle v\cdot \psi =2^{\frac {1}{2}}(w\wedge \psi +\iota (w^{*})\psi ),}$

where the second term is a contraction (interior multiplication) defined using the bilinear form, which pairs W and W^∗. This action respects the Clifford relations v² = Q(v)1, and so induces a homomorphism from the Clifford algebra Cl_nC of V to End(S). A similar action can be defined on S′, so that both S and S′ are Clifford modules.

The Lie algebra so(n, C) is isomorphic to the complexified Lie algebra spin_n^C in Cl_nC via the mapping induced by the covering Spin(n) → SO(n)^[2]

${\displaystyle v\wedge w\mapsto {\tfrac {1}{4}}[v,w].}$

It follows that both S and S′ are representations of so(n, C). They are actually equivalent representations, so we focus on S.

The explicit description shows that the elements α_i ∧ a_i of the Cartan subalgebra h act on S by

${\displaystyle (\alpha _{i}\wedge a_{i})\cdot \psi ={\tfrac {1}{4}}(2^{\tfrac {1}{2}})^{2}(\iota (\alpha _{i})(a_{i}\wedge \psi )-a_{i}\wedge (\iota (\alpha _{i})\psi ))={\tfrac {1}{2}}\psi -a_{i}\wedge (\iota (\alpha _{i})\psi ).}$

A basis for S is given by elements of the form

${\displaystyle a_{i_{1}}\wedge a_{i_{2}}\wedge \cdots \wedge a_{i_{k}}}$

for 0 ≤ k ≤ m and i₁ < ... < i_k. These clearly span weight spaces for the action of h: α_i ∧ a_i has eigenvalue −1/2 on the given basis vector if i = i_j for some j, and has eigenvalue 1/2 otherwise.

It follows that the weights of S are all possible combinations of

${\displaystyle {\bigl (}\pm {\tfrac {1}{2}},\pm {\tfrac {1}{2}},\ldots \pm {\tfrac {1}{2}}{\bigr )}}$

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

When n is even, S is not an irreducible representation: ${\displaystyle S_{+}=\wedge ^{\mathrm {even} }W}$ and ${\displaystyle S_{-}=\wedge ^{\mathrm {odd} }W}$ 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 2^m−1 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

${\displaystyle {\bigl (}{\tfrac {1}{2}},{\tfrac {1}{2}},\ldots {\tfrac {1}{2}},{\tfrac {1}{2}}{\bigr )}}$ and ${\displaystyle {\bigl (}{\tfrac {1}{2}},{\tfrac {1}{2}},\ldots {\tfrac {1}{2}},-{\tfrac {1}{2}}{\bigr )}}$

respectively. The Clifford action identifies Cl_nC 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 2^m: the Clifford action of a unit vector u ∈ U is given by

${\displaystyle u\cdot \psi =\left\{{\begin{matrix}\psi &{\hbox{if }}\psi \in \wedge ^{\mathrm {even} }W\\-\psi &{\hbox{if }}\psi \in \wedge ^{\mathrm {odd} }W\end{matrix}}\right.}$

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

${\displaystyle {\bigl (}{\tfrac {1}{2}},{\tfrac {1}{2}},\ldots {\tfrac {1}{2}}{\bigr )}.}$

The Clifford action is not faithful on S: Cl_nC 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 α of Cl_nC (also known as the principal automorphism), which is the identity on the even subalgebra, and minus the identity on the odd part of Cl_nC. In other words, there is a linear isomorphism from S to S′, which identifies the action of A in Cl_nC on S with the action of α(A) on S′.

Bilinear forms

if λ is a weight of S, so is −λ. 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 β on S via

${\displaystyle \beta (\varphi ,\psi )=B(\varphi )(\psi ).}$

Here invariance means that

${\displaystyle \beta (\xi \cdot \varphi ,\psi )+\beta (\varphi ,\xi \cdot \psi )=0}$

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

${\displaystyle \quad \beta (A\cdot \varphi ,\psi )=\beta (\varphi ,\tau (A)\cdot \psi )\qquad (1)}$

for any A in Cl_nC. In fact τ 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 α, which is the automorphism induced by minus the identity on V. Both satisfy τ(ξ) = −ξ for ξ in so(n,C).

When n = 2m, the situation depends more sensitively upon the parity of m. For m even, a weight λ has an even number of minus signs if and only if −λ 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, λ is a weight of S₊ if and only if −λ 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 β corresponding to B satisfies (1), where τ is a fixed antiautomorphism (either reversion or conjugation).

Symmetry and the tensor square

The symmetry properties of β: 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 β(φ,ψ) = ε β(ψ,φ) and τ has sign ε_k on ∧^kV then

${\displaystyle \beta (A\cdot \varphi ,\psi )=\varepsilon \varepsilon _{k}\beta (A\cdot \psi ,\varphi )}$

for A in ∧^kV.

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

${\displaystyle S\otimes S\cong \bigoplus _{j=0}^{m}\wedge ^{2j}V^{*}}$

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

${\displaystyle \sum _{j=0}^{m}(-1)^{j}\dim \wedge ^{2j}\mathbb {C} ^{2m+1}=(-1)^{{\frac {1}{2}}m(m+1)}2^{m}=(-1)^{{\frac {1}{2}}m(m+1)}(\dim \mathrm {S} ^{2}S-\dim \wedge ^{2}S)}$

and the sign determines which representations occur in S²S and which occur in ∧²S.^[3] In particular

${\displaystyle \beta (\phi ,\psi )=(-1)^{{\frac {1}{2}}m(m+1)}\beta (\psi ,\phi ),}$ and

${\displaystyle \beta (v\cdot \phi ,\psi )=(-1)^{m}(-1)^{{\frac {1}{2}}m(m+1)}\beta (v\cdot \psi ,\phi )=(-1)^{m}\beta (\phi ,v\cdot \psi )}$

for v ∈ V (which is isomorphic to ∧^2mV), confirming that τ 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:

n mod 8	0	1	2	3	4	5	6	7
Spinor algebra	${\displaystyle {\mathfrak {so}}(S_{+})\oplus {\mathfrak {so}}(S_{-})}$	${\displaystyle {\mathfrak {so}}(S)}$	${\displaystyle {\mathfrak {gl}}(S_{\pm })}$	${\displaystyle {\mathfrak {sp}}(S)}$	${\displaystyle {\mathfrak {sp}}(S_{+})\oplus {\mathfrak {sp}}(S_{-})}$	${\displaystyle {\mathfrak {sp}}(S)}$	${\displaystyle {\mathfrak {gl}}(S_{\pm })}$	${\displaystyle {\mathfrak {so}}(S)}$

For n ≤ 6, these embeddings are isomorphisms (onto sl rather than gl for n = 6):

${\displaystyle {\mathfrak {so}}(2,\mathbb {C} )\cong {\mathfrak {gl}}(1,\mathbb {C} )\qquad (=\mathbb {C} )}$

${\displaystyle {\mathfrak {so}}(3,\mathbb {C} )\cong {\mathfrak {sp}}(2,\mathbb {C} )\qquad (={\mathfrak {sl}}(2,\mathbb {C} ))}$

${\displaystyle {\mathfrak {so}}(4,\mathbb {C} )\cong {\mathfrak {sp}}(2,\mathbb {C} )\oplus {\mathfrak {sp}}(2,\mathbb {C} )}$

${\displaystyle {\mathfrak {so}}(5,\mathbb {C} )\cong {\mathfrak {sp}}(4,\mathbb {C} )}$

${\displaystyle {\mathfrak {so}}(6,\mathbb {C} )\cong {\mathfrak {sl}}(4,\mathbb {C} ).}$

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.

(†) N is even for n > 3 and for n = 3, this is sp(1).

The even-dimensional case is similar. For n > 2, the complex half-spin representations are even-dimensional. We have additionally to deal with hermitian structures and the real forms of sl(2N, C), which are sl(2N, R), su(K, L) with K + L = 2N, and sl(N, H). The resulting even spin representations are summarized as follows.

(*) For pq = 0, we have instead so(2N) + so(2N)

(†) N is even for n > 4 and for pq = 0 (which includes n = 4 with N = 1), we have instead sp(N) + sp(N)

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

Euclidean signature	Minkowskian signature	Other signatures
${\displaystyle {\mathfrak {so}}(2)\cong {\mathfrak {u}}(1)}$	${\displaystyle {\mathfrak {so}}(1,1)\cong \mathbb {R} }$
${\displaystyle {\mathfrak {so}}(3)\cong {\mathfrak {sp}}(1)}$	${\displaystyle {\mathfrak {so}}(2,1)\cong {\mathfrak {sl}}(2,\mathbb {R} )}$
${\displaystyle {\mathfrak {so}}(4)\cong {\mathfrak {sp}}(1)\oplus {\mathfrak {sp}}(1)}$	${\displaystyle {\mathfrak {so}}(3,1)\cong {\mathfrak {sl}}(2,\mathbb {C} )}$	${\displaystyle {\mathfrak {so}}(2,2)\cong {\mathfrak {sl}}(2,\mathbb {R} )\oplus {\mathfrak {sl}}(2,\mathbb {R} )}$
${\displaystyle {\mathfrak {so}}(5)\cong {\mathfrak {sp}}(2)}$	${\displaystyle {\mathfrak {so}}(4,1)\cong {\mathfrak {sp}}(1,1)}$	${\displaystyle {\mathfrak {so}}(3,2)\cong {\mathfrak {sp}}(4,\mathbb {R} )}$
${\displaystyle {\mathfrak {so}}(6)\cong {\mathfrak {su}}(4)}$	${\displaystyle {\mathfrak {so}}(5,1)\cong {\mathfrak {sl}}(2,\mathbb {H} )}$	${\displaystyle {\mathfrak {so}}(4,2)\cong {\mathfrak {su}}(2,2)}$	${\displaystyle {\mathfrak {so}}(3,3)\cong {\mathfrak {sl}}(4,\mathbb {R} )}$

The only special isomorphisms of real Lie algebras missing from this table are ${\displaystyle {\mathfrak {so}}^{*}(3,\mathbb {H} )\cong {\mathfrak {su}}(3,1)}$ and ${\displaystyle {\mathfrak {so}}^{*}(4,\mathbb {H} )\cong {\mathfrak {so}}(6,2).}$