The Inner Poduct Space Definition. Suppose V is a vector space 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 bilinear form 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 inner product space 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 isometry 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 .