The Inner Poduct Space Definition. Suppose V is a over F ≡ R, C or H and ε : V × V → F is a biadditive map; i.e. ε(x + y, z)=ε(x, z)+ε(y, z), and ε(z,x + y)=ε(z,x)+ε(z,y), ∀x, y, z ∈ V. The biadditive map ε is said to be (pure) bilinear if ε(λx, y)=λε(x, y), and ε(x, λy)=λε(x, y) ∀λ ∈ F. The biadditive map ε is said to be (hermitian) bilinear if ε(xλ, y)=λε(x, y), and ε(x, yλ)=λε(x, y) ∀λ ∈ F. • If F ≡ H and ε is pure bilinear, then ε = 0 because ε(x, y)λµ = ε(xλ, yµ)=ε(x, y)µλ, while µ, λ ∈ H can be chosen so that [µ, λ] ≡ µλ − λµ =0.6 Thus, for F ≡ H there is only one kind of bilinear, namely, hermitian bilinear. Definition. Suppose V is a finite dimensional vector space over F ≡ R, C or H. An inner product ε on V is a nondegenerate on V that is either symmetric or skew. If F = C, then there are two types of symmetric and two types of skew: pure and hermitian. The four types of inner products are: (1) R-symmetric: ε is R-bilinear and ε(x, y)=ε(y, x), (2) C-symmetric: ε is C-bilinear and ε(x, y)=ε(y, x), (3) C-hermitian symmetric: ε is C-hermitian bilinear and ε(x, y)=ε(y, x), (4) H-hermitian symmetric: ε is H-hermitian bilinear and ε(x, y)=ε(y, x). The four types of inner products are: (1) R-skew or R-sympletic: ε is R-bilinear and ε(x, y)=−ε(y, x), (2) C-skew or C-sympletic: ε is C-bilinear and ε(x, y)=−ε(y, x), (3) C-hermitian skew: ε is C-hermitian bilinear and ε(x, y)=−ε(y, x), (4) H-hermitian skew: ε is H-hermitian bilinear and ε(x, y)=−ε(y, x).

The Standard Models (1) R-symmetric: The vector space is Rn, denoted by R(p, q), p + q = n, with

ε(x, y)=x1y1 + ···+ xpyp −···−xnyn; n (2) C-symmetric: C with ε(z,w)=z1w1 + ···+ znwn; (3) C-hermitian symmetric: The vector space is Cn, denoted by C(p, q), p+q = n, with ε(z,w)=z1w1 + ···+ zpwp −···−znwn; (4) H-hermitian symmetric: The vector space is Hn, denoted by H(p, q), p+q = n, with ε(x, y)=x1y1 + ···+ xpyp −···−xnyn.

Typeset by AMS-TEX

1 2

And (1) R-skew or R-sympletic: The vector space is R2n with

ε(x, y)=x1y2 − x2y1 + ···+ x2n−1y2n −···−x2ny2n−1;

ε =dx1 ∧ dx2 + ···+ dx2n−1 ∧ dx2n;

(2) C-skew or C-sympletic: The vector space is C2n with

ε(z,w)=z1w2 − z2w1 + ···+ z2n−1w2n − z2nw2n−1;

ε =dz1 ∧ dz2 + ···+ dz2n−1 ∧ dz2n;

(3) C-hermitian skew: The vector space is Cn, denoted by C(p, q), p + q = n, with ε(z,w)=iz1w1 + ···+ izpwp −···−iznwn; (4) H-hermitian skew: The vector space is Hn with

ε(x, y)=x1iy1 + ···+ xniyn.

Basic Theorem 5. Suppose (V,ε) is an of one of the eight types. Then V is isometric to the standard model of the same type that has the same dimension and signature.

Corollary 6. Suppose (V,ε) and (V,ε) are two inner product spaces of the same e type. The dimension and signature aree the same iff V and V are isometric. e 3

The Parts of An Inner Product • Suppose ε is one of the eight types of inner products. If ε is either C-valued or H-valued, then ε has various parts. (I) The simplest case is when ε is a complex-valued inner product. Then ε has a real part α and an imaginary part β defined by the equation ε = α + iβ. (II) Suppose ε is a quaterion valued inner product. There are several options for analyzing the part of ε. (II.1) First, ε has a real part α and a pure imaginary part β defined by ε = α + β where α =Reε is real-valued and β =Imε takes on values in Im H = span{i, j, k}. – The imaginary part β has three components defined by

β = iβ1 + jβ2 + kβ3,

where the parts β1, β2, β3 are real-valued.

(II.2) Second, using the complex-valued Ri (right multiplication by i)onH, each quaterion x ∈ H has a unique decomposition x = z + jw, where z, w ∈ C. Therefore ε = γ + jδ where γ = α + iβi and δ = βj − iβk.

(I) C-Hermitian Symmetric • Consider the standard C-hermitian (symmetric) form

ε(z,w)=z1w1 + ···+ zpwp −···−znwn. with signature p, q on Cn. Since ε is complex-valued, it has a real and imaginary part given by ε = g − iw. For z = x+iy and w = ξ +iη ∈ Cn =∼ R2n, the real and imaginary parts g =Reε and ω = −Im ε are given by

g(z,w)=x1ξ1 + y1η1 + ···+ xpξp + ypηp −···−xnξn − ynηn and ω(z,w)=−x1η1 + y1ξ1 −···+ xnηn − ynξn. • Thus, g is the standard R-symmetric form on R2n with signature 2p,2q. • Modulo some sign changes, w is the standard sympletic form on R2n. In this context, when q =0,w is exactly the standard K¨ahlerform on Cn and is usually written as i i w = dz ∧ dz + ···+ dz ∧ dz . 2 1 1 2 n n 4

Lemma 7. Suppose ε is C-hermitian symmetric (signature p, q) on a complex vector space V with complex structure i. Then g =Reε is R-symmetric with signature 2p, 2q and w =Imε is R-skew. Moreover, each determines the other by

(1) g(z,w)=ω(iz, w) and ω(z,w)=g(iz, w).

Also i is an for both g and w:

g(iz, iw)=g(z,w) and ω(iz, iw)=ω(z,w).

• Conversely, given R-symmetric form g with i an isometry, if ω is determined by (1), then ε = g − iω is C-hermitian. Also, given R-skew form ω with i an isometry, if g is determined by (1), then ε = g − iω is C-hermitian.

Remark. Lemma 7 can be summarized by saying that . “The confluence of any two of (a) complex geometry, (b) sympletic geometry, (c) Riemannian geometry is K¨ahlergeomety.”

Definition. Suppose (V,ε) is C-hermitian symmetric inner product space with signature p, q. Let

GL(n, C) =EndC(V ), U(p, q) =the subgroup of GL(V,C) fixing ε, O(2p, 2q) =the subgroup of GL(V,R) fixing g =Reε, Spin(V,R) =the subgroup of GL(V,R) fixing ω =Imε.

Corollary 8. The intersection of any two of the three groups

GL(n, C), Spin(n, R), and O(2p, 2q)

is the group U(p, q). 5

(II) H-Hermitian Symmetric • Consider the standard C-hermitian (symmetric) form

ε(z,w)=z1w1 + ···+ zpwp −···−znwn.

t with signature p, q on Cn. Note that ε is H-valued. – As noted earlier, it is natural to consider H as two copies of C,

H =∼ C ⊕ jC, or z = z + jw.

In particular, ε = h + jσ, with h and σ complex-valued.

• For x = z + jw and y = ξ + jη with z,w,ξ,η ∈ C,

xy =(z − jw)(ξ + jη)=zξ − jwjη + zjη − jwξ (3) =zξ + wη + j(zη − wξ).

Therefore, the first complex part of h of ε is given by

h(x, y)=z1ξ1 + w1η1 + ···+ zpξp + wpηp −···−znξn − wnηn

Thus h is the standard C-hermitian symmetric form on C2n =∼ Hn. • Because of (3) the second complex part σ is given by

σ(x, y)=z1η1 − w1ξ1 ±···±znηn ± wnξn.

• Thus, modulo some sign changes, w is the standard C-skew form on C2n. Lemma 9. Suppose ε = h + jσ is H-hermitian symmetric on a right H-space V . Then the first complex part h of ε is C-hermitian symmetric and the second complex part of ε is C-skew. Moreover, each determines the other by

(4) h(x, y)=σ(x, yj) and σ(x, y)=−h(x, yj).

Also

(5) h(xj, yj)=h(x, y) and σ(xj, yj)=σ(x, y).

• Conversely, given C-hermitian symmetric form h with h(xj, yj)=h(x, y),ifσ is determined by (4), then ε = h + jσ is H-hermitian symmetric. Also, given C-skew form σ with σ(xj, yj)=σ(x, y),ifh is determined by (4), then ε = h + iσ is C-hermitian. Proof. The identity ε(x, yj)=ε(x, y)j can be used to prove (4), while ε(xj, yj)= −jε(x, y)j can be used to prove (5).  6

Definition. Suppose (V,ε) is C-hermitian symmetric inner product space with signature p, q. Let

GL(n, H) =EndH(V ), HU(p, q) =the subgroup of GL(V,H) fixing ε, U(2p, 2q) =the subgroup of GL(V,C) fixing the first complex part h of ε, Spin(V,C) =the subgroup of GL(V,C) fixing the second complex part σ of ε.

Corollary 10. The intersection of any two of the three groups

GL(n, H), Spin(n, C), and U(2p, 2q) is the group HU(p, q).

• It is useful to construct the quaterionic structure from h and σ. • Suppose V is a complex 2n-dimensional vector space, h is a C-hermitian symmetric inner product on V , and σ is a complex sympletic inner product on V . Then h and σ define a complex antilinear map J by

h(xJ, y)=σ(x, y).

Now

σ(xJ 2,y)=−σ(y, xJ2)=−h(yJ, xJ2)=−h(xJ 2,yJ)=−σ(xJ, yJ).

Therefore J 2 = −1iff σ(xJ, yJ)=σ(x, y). In this case, h and σ are said to be compatible. Lemma 10. Suppose V is a complex 2n-dimensional vector space, h is a C- hermitian symmetric inner product on V , and σ is a complex sympletic inner prod- uct on V .Ifh and σ are compatible, then they determine a right H-structure on V and ε def= h + jσ is an H-hermitian symmetric inner product on V . Proof. It remains to verify that h + jσ is H-hermitian symmetric.  7

The Quaterion Vector Space Hn • The quaterion vector space Hn can be considered as complex vector space in a varierty of natural ways (more pecisely, a 2-sphere S2 of natural ways.) • Let Im H denote the real hyperplane in H with normal 1 ∈ H. Let S2 denote the unit sphere in Im H. – Then, for each u ∈ S2, u2 = −uu = −|u|2 = −1. Therefore, right multiplication by u, defined by

n Rux ≡ xu ∀x ∈ H .

n 2 is a complex structure on H ; that is, Ru = −1. – This property enables one to define a complex scalar multiplication on Hn by

n 2 (a + bi)x ≡ (a + bRu)(x), ∀a, b ∈ R,x∈ H where i = −1.

n n n • Note that EndH(H ) ⊂EndC(H ) for each of the complex structure Ru on H , where u ∈ S2 ⊂ ImH. • Choosing a complex basis for Hn provides a complex linear isomorphism

Hn =∼ C2n.

— Sometimes it is convenient to select the complex basis as follows. – Let C(u) denote the complex line containing 1 in each of the axis subspaces H ⊂ Hn. Thus, C(u) is the real span of 1 and u. – Let C(u)⊥ denote the complex line orthogonal to C(u)inH ⊂ Hn. Then

Hn =∼ [C(u) ⊕ C(u)⊥]n =∼ C2n.

Another Description of HU(p, q).

• Recall that for each u ∈Im H, right multiplication by u (denoted by Ru) acting on Hn determines a complex structure on Hn and hence an isomorphism Hn =∼ C2n. Each of the complex structures Ru determines a K¨ahlerform

ωu(x, y)=Reε(xu, y).

Let g(x, y)=Reε(x, y). Then, we have the following Lemma 11. For all x, y ∈ Hn,

ε = g + iωi + jωj + kωk.

Proof. For u ∈ Im H with |u|2 = uu = −u2 =1,

hu, ε(x, y)i = h1, uε(x, y)i =Reε(xu, y)=ωu(x, y).  8

Corollary 12. The hyper-unitary group HU(p, q) is the intersection of the three n unitary groups determined by the three complex structures Ri, Rj , Rk on H . • For each complex structure u ∈Im H, |u| = 1, the complex C(u) valued form

hu = g + uωu

is C(u)-Hermitian symmetric. – The group that fixes hu is a unitary group with signature 2p,2q determined by the complex structure Ru. • HU(p, q) is also the intersection, over u ∈ S2 ⊂ Im H, of the unitary groups determined by all the complex structures Ru. – The simplest case states that HU(1) is the intersection of all unitary groups ∼ 2 determined by the complex structures Ru on H = C .