Download The Inner Poduct Space Definition. Suppose V is a vector space

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 6= 0.
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) = x1 y1 + · · · + xp yp − · · · − xn yn ;
(2) C-symmetric: Cn with ε(z, w) = z1 w1 + · · · + zn wn ;
(3) C-hermitian symmetric: The vector space is Cn , denoted by C(p, q), p + q =
n, with
ε(z, w) = z 1 w1 + · · · + z p wp − · · · − z n wn ;
(4) H-hermitian symmetric: The vector space is Hn , denoted by H(p, q), p+q =
n, with
ε(x, y) = x1 y1 + · · · + xp yp − · · · − xn yn .
Typeset by AMS-TEX
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And
(1) R-skew or R-sympletic: The vector space is R2n with
ε(x, y) =x1 y2 − x2 y1 + · · · + x2n−1 y2n − · · · − x2n y2n−1 ;
ε =dx1 ∧ dx2 + · · · + dx2n−1 ∧ dx2n ;
(2) C-skew or C-sympletic: The vector space is C2n with
ε(z, w) =z1 w2 − z2 w1 + · · · + z2n−1 w2n − z2n w2n−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) = iz 1 w1 + · · · + iz p wp − · · · − iz n wn ;
(4) H-hermitian skew: The vector space is Hn with
ε(x, y) = x1 iy1 + · · · + xn iyn .
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 (Ve , εe) are two inner product spaces of the same
type. The dimension and signature are the same iff V and Ve are isometric.
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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 = Re ε is real-valued and β = Im ε 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) on H,
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) = z 1 w1 + · · · + z p wp − · · · − z n wn .
with signature p, q on Cn . Since ε is complex-valued, it has a real and imaginary
part given by
ε = g − iw.
n ∼ 2n
For z = x+iy and w = ξ +iη ∈ C = R , the real and imaginary parts g = Re ε
and ω = −Im ε are given by
g(z, w) = x1 ξ1 + y1 η1 + · · · + xp ξp + yp ηp − · · · − xn ξn − yn ηn
and
ω(z, w) = −x1 η1 + y1 ξ1 − · · · + xn ηn − yn ξn .
• 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 Kähler form on Cn
and is usually written as
i
i
w = dz1 ∧ dz1 + · · · + dzn ∧ dzn .
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Lemma 7. Suppose ε is C-hermitian symmetric (signature p, q) on a complex
vector space V with complex structure i.
Then g = Re ε is R-symmetric with signature 2p, 2q and w = Im ε 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 Kähler geomety.”
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 = Re ε,
Spin(V, R) =the subgroup of GL(V, R) fixing ω = Im ε.
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).
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(II) H-Hermitian Symmetric
• Consider the standard C-hermitian (symmetric) form
ε(z, w) = z 1 w1 + · · · + zp wp − · · · − z n wn .
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) = z 1 ξ1 + w 1 η1 + · · · + z p ξp + w p ηp − · · · − z n ξn − w n ηn
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) = z1 η1 − w1 ξ1 ± · · · ± zn ηn ± wn ξn .
• 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), if σ 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), if h 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, xJ 2 ) = −h(yJ, xJ 2 ) = −h(xJ 2 , yJ) = −σ(xJ, yJ).
Therefore J 2 = −1 iff
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 Chermitian symmetric inner product on V , and σ is a complex sympletic inner product on V . If h and σ are compatible, then they determine a right H-structure on
V and
def
ε = h + jσ
is an H-hermitian symmetric inner product on V .
Proof. It remains to verify that h + jσ is H-hermitian symmetric.
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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
Ru x ≡ xu ∀x ∈ Hn .
is a complex structure on Hn ; that is, Ru2 = −1.
– This property enables one to define a complex scalar multiplication on Hn by
(a + bi)x ≡ (a + bRu )(x), ∀a, b ∈ R, x ∈ Hn where i2 = −1.
• Note that EndH (Hn ) ⊂EndC (Hn ) for each of the complex structure Ru on Hn ,
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) in H ⊂ 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 Kähler form
ωu (x, y) = Re ε(xu, y).
Let g(x, y) = Re ε(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 = Re ε(xu, y) = ωu (x, y).
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Corollary 12. The hyper-unitary group HU(p, q) is the intersection of the three
unitary groups determined by the three complex structures Ri , Rj , Rk on Hn .
• 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
determined by the complex structures Ru on H ∼
= C2 .