FCCR Table of Contents
This appendix contains definitions and notational conventions and results regarding G(n)-Hermiticity also related to those pertaining to the invariance groups of G(n) [Section XI], as well as a very condensed review of some relevant aspects of complex manifolds and associated Lie groups, [Appendix B], [Chevalley 1946], [Pontryagin 1966], [Wolf 1967], [Kobayashi 1969], [Helgason 1962], [Bourbaki 1966].
Consider the Hilbert space Hilb(n) [Appendix A] and its canonical basis |n, k>. The operator G(n) of [Section II] defines an indefinite Hermitean form on Hilb(n), which then technically becomes a pseudohilbert space. [Nevertheless, we will continue for brevity to call it a Hilbert space.]
h(x, y) := <x|G(n)|y> (D.1)
For A ∈ Alg(H(n)), the C*-algebra of
linear operators on Hilb(n), A is:
left G(n)-skew (LGS), iff
G(n) A = -A† G(n)
where † indicates Hermitean conjugation (transpose and complex
conjugation);
left G(n)-Hermitean (LGH), iff
G(n) A = A† G(n)
right G(n)-skew (RGS), iff
A G(n) = -G(n) A†
right G(n)-Hermitean (RGH), iff
A G(n) = G(n) A†
Let U(G, n), the Adjoint Group,
[Section XI]
be the set of transformations V, in GL(n, C)
that preserve G(n) in their left action,
V† G(n) V = G(n)
written set theoretically as
U†(G, n) G(n) U(G, n) = G(n)
and similarly define the Coadjoint Group, so the elements of
U(G-1, n) preserve G(n) in a right action,
V G(n) V† = G(n)
written set theoretically as
U(G-1, n) G(n) U†(G-1, n) = G(n)
or
U†(G-1, n) G-1(n) U(G-1, n) = G-1(n)
On the |n, k> eigenbasis, define square roots of G(n),
G±1/2(n) := Diag[ I(n-1), ±i√(n-1) ]
Then the following relations hold:
G±1/2(n) G∓-1/2(n) = I(n)
G±1/2(n) G±-1/2(n) = γ(n)
G±1/2(n) G±1/2(n) = G(n)
G±1/2(n) G∓1/2(n) = γ(n)G(n)
G±1/2†(n) γ(n) G±1/2(n) = G(n)
G±1/2†(n) = G∓1/2(n)
G±-1/2†(n) = G∓-1/2(n)
[G±1/2(n), G∓1/2(n)] = 0
G(n) G±-1/2(n) = G∓1/2(n)
G-1(n) G±1/2(n) = G∓-1/2(n)
γ(n) G±1/2(n) = G∓1/2(n)
γ(n) G±-1/2(n) = G∓-1/2(n)
where γ(n) is defined, again on the canonical basis, by its
matrix elements
γ(n) := Diag[ I(n-1), -1]
γ(n) is the form whose preservation defines U(n-1, 1).
Note that the ± tags simply reflect the sign of the one
imaginary component. This perhaps unfortunately confusing notation
was instituted because I was reluctant to make an absolute sign choice
regarding the "coherent square roots".
[An nxn unit matrix 2n independent square roots.]
Some of the propositions that follow simply need to be stated.
Their proofs are elementary or so obvious as to arise by inspection,
and so they are not proved.
G(n) is both LGH and RGH, while iG(n) is both LGS and RGS.
While G±1/2(n) and G±-1/2(n)
are neither LGH nor RGH,
G±+1/2(n) + G∓+1/2(n) is both LGH and RGH
G±-1/2(n) + G∓-1/2(n) is both LGH and RGH
and
G±+1/2(n) - G∓+1/2(n) is both LGS and RGS
G±-1/2(n) - G∓-1/2(n) is both LGS and RGS
If A is Hermitean in the usual sense, i.e., A† = A, then,
by similarity transformations,
±A := G±-1/2(n) A G∓+1/2(n)
is LGH, and
A± := G±+1/2(n) A G∓-1/2(n)
is RGH
Similarly if A is skewhermitean in the usual sense, A† = -A, then,
±A := G±-1/2(n) A G∓+1/2(n)
is LGS, and
A± := G±+1/2(n) A G∓-1/2(n)
is RGS.
The adjoint group U(G, n) and the coadjoint group
U(G-1, n)
are semisimple Lie subgroups of GL(n, C) that are
conjugate to each other by the transformation
G U(G, n) G-1 = U(G-1, n)
The Lie algebra u(G, n) of U(G, n) is the set of LGS matrices
in the Lie algebra gl(n, C) and the Lie algebra
u(G-1, n) of U(G-1, n) is the set of
RGS matrices in gl(n, C). Both U(G, n) and U(G-1, n)
are conjugate in GL(n, C) to the pseudounitary group U(n-1, 1).
Then also, both u(G, n) and u(G-1, n) are
conjugate to the Lie algebra u(n-1, 1).
The conjugations from the pseudounitary group and
algebra to the adjoint and coadjoint groups and algebras are given
by the square root transformations given in
[definition D.3].
Both u(G, n) and u(G-1, n) are conjugations
[definition D.4].
of u(n-1, 1).
If A and B are any two left/right skew matrices, then
[A, B] is left/right G(n)-skew
iA is left/right G(n)-Hermitean
The maximal compact subgroups of the groups SU(n-1, 1), SU(G, n)
and SU(G-1, n) are isomorphic by group conjugation
in GL(n, C) to
S(U(n) X U(1))
and the Cartan canonical decomposition
[Appendix B]
of the Lie algebra su(n-1, 1) can be written
| | | | | |
| h | 0 | | 0 | |a>|
| | | + | | |
| | | | | |
|-----------|----| |-----------|----|
| 0 | iw | | <a| | 0 |
where h is Hermitean and w is real. The first matrix
is an element of the maximal compact subgroup.
The non-Hermitean contribution the first matrix is
from the single element "iw".
The second matrix
is an element of the invariant subspace.
The vector <a|
is the complex conjugate transpose of |a>.
making the second matrix Hermitean.
From the conjugation mappings the adjoint algebra u(G, n)
has the decomposition
| | | | | |
| h | 0 | | 0 | |a>√(n-1) |
| | | + | | |
| | | | | |
|-----------|----| |-----------|-----------|
| 0 | iw | | <a|√(n-1) | 0 |
The matrix sum is, of course LGS while the decomposition of the
coadjoint algebra u(G-1, n) of RGS matrices is
| | | | | |
| h | 0 | | 0 | |a>√(n-1) |
| | | + | | |
| | | | | |
|-----------|----| |-----------|-----------|
| 0 | iw | |√(n-1) <a| | 0 |
Consistent with the definitions of left/right G(n)-hermiticity
and G(n)-skewness,
define the operations of right G(n)-conjugation:
A -> AG := G(n) A† G-1(n)
and left G(n)-conjugation
A -> GA := G-1(n) A† G(n)
<Proposition D.6>:
[Corrected: December 18, 2004]
Assume arbitrary A and B.
As (AB)† = B† A†,
(AB)G = BG AG
G(AB) = GB GA
Regarding the trace and determinant functionals,
Tr( AG ) = Tr( A† ) = Tr( A )*
Det( AG ) = Det( A† ) = Det( A )*
The operations of right and left G(n)-conjugation are each involutory,
(AG)G = A, G(GA) = A
but, they are not commuting operations, meaning that their notation does
not associate,
(GA)G = G2 A G-2
G(AG) = G-2 A G2
Then, as A may be split into a sum of Hermiteam and Skewhermitean parts
A = (1/2)(A + A†) + (1/2)(A - A†)
the Hermiticity and Skewhermiticity of the terms being invariant under
unitary transformations, so also by,
A = (1/2)(A + AG) + (1/2)(A - AG)
A = (1/2)(A + GA) + (1/2)(A - GA)
can A be split regarding R and L G-hermitean and skew terms that
keep these properties under the adjoint actions of U(G, n) and
U(G-1, n) respectively.
Also, the G-conjugation operations do not commute with the complex
conjugate transpose operation
(A†)G = G(n) A G-1(n) = (GA)†
(AG)† = G-1(n) A G(n) = G(A†)
An operator is left (right) G(n)-Hermitean if it is invariant
under left (right) G(n)-conjugation.
G(n) is not an element of U(G, n), while it is
obviously an element of the algebra u(G, n) and of
u(G-1, n), the G(n)-conjugate algebra,
since it is invariant under both conjugations.
The element,
exp( i θ G(n) )
is of course, a one-parameter subgroup of both U(G, n)
and U(G-1, n).
There is nothing difficult or tricky about the proofs.
If A is arbitrary, the quantities
(A AG) and (AG A)
are right G(n)-Hermitean and,
(A GA) and (GA A)
are left G(n)-Hermitean.
If h is right G(n)-Hermitean, then
s = ih
is right G(n)-skew and with
u := exp( s )
u† = G-1(n) exp( -s ) G(n)
uG = exp( -s ) = u-1
u G(n) u† = G(n)
uG† G-1(n) uG = G-1(n)
u = exp( s ) ∈ U(G, n)
If U ∈ U(G, n),
its left adjoint action on u(G, n),
is an automorphism of u(G, n), preserving left G(n)-skewness
and left G(n)-hermiticity;
similarly if U ∈ U(G-1, n),
its right adjoint action on u(G-1, n),
is an automorphism of u(G-1, n),
preserving right G(n)-skewness and right G(n)-hermiticity.
Proof:
Let A be an element of u(G, n), it is left G(n)-skew, and let U be
an element of U(G, n).
U† G(n) U = G(n)
G(n) A = - A† G(n)
then
U† G(n) U U-1 A U = -U† A† G(n) U
= -U† A† (U†)-1 U† G(n) U
G(n) (U-1 A U) = -U† A† (U-1)† G(n)
= -(U-1 A U)† G(n)
where
A -> U-1 A U
is the adjoint action of U(G, n) on u(G, n). The proof for the
right counterpart as well as the argument for right G(n)-hermiticity,
proceeds in like manner.
QED
Any left (right) G(n)-Hermitean operator, is diagonalizable by
an adjoint action of U(G, n) (U(G-1, n)).
Proof:
From
proposition D.2,
there is a non-singular map from
Euclidean Hermitean operators
to (left) right G(n)-Hermitean operators, and similarly from unitary
operators to right (left) G(n)-pseudounitary operators. Since
any Euclidean Hermitean operator can be diagonalized by a unitary
operator.
U-1 H U = D
U-1 G∓-1/2 G±+1/2 H G∓-1/2 G±+1/2 U = D
G±+1/2 U-1 G∓-1/2 G±+1/2 H G∓-1/2 G±+1/2 U G∓-1/2
= G±+1/2 D G∓-1/2
For U unitary and H Euclidean Hermitean, let
V = (G±+1/2 U G∓-1/2)
K = (G±+1/2 H G∓-1/2)
any V and any K is so representable.
Then V ∈ U(G, n) and K ∈ u(G, n), and
V-1 K V = D
where D is diagonal.
The argument proceeds in the same way for the left statement.
QED
Note that the diagonal form D is achieved by a mapping of the same
structure as for the Hermitean-unitary problem. It is a matter of
the adjoint action of a Lie group on its Lie algebra.
U-1 H U = V-1 K V = D
Polar Decomposition and its Analogs
The separation of an operator into a product of a unitary and
Hermitean operators is not preserved under U(G, n).
It is of interest to find the analogous decomposition of an
operator into the product of a right G(n)-Hermitean operator
and an element of U(G, n). A theorem for
the usual polar decomposition is:
For any nxn nonsingular matrix A, there are unique polar decompositions
A = UH' = HU
where U is unitary and H is Euclidean Hermitean and
H' = U-1 H U
and so H' is also Euclidean Hermitean.
Proof:
Let
H = (AA†)1/2
and
U = (AA†)-1/2 A
Simply verify the construction. Cf. [Gel'fand 1961], p. 111.
QED
Now again let A be any non-singular operator. We know of the
decomposition
A = H U
where H is Hermitean and U is unitary exists.
Any similarity map of any A in GL(n, C)
A -> S A S-1
can be understood as simply a change of basis, or an inner
automorphism of GL(n, C), with A -> A'. Then
±A' := G±-1/2(n) A G∓+1/2(n)
±A' = G±-1/2(n) H G∓+1/2(n) G±-1/2(n) U G∓+1/2(n)
for either sign expresses A' as "adjoint" decomposition into an element
of U(G, n) and a right G(n)-Hermitean operator.
A similar "coadjoint" decomposition can also be constructed into an
element of U(G-1, n) and a left G(n)-Hermitean operator.
Eine Kleine Liegruppentheorie
It is known that the pseudounitary Lie group U(n-1, 1) is semisimple,
connected and noncompact;
that as compact closed subgroups, O(n-1) the orthogonal group
and U(n-1) the unitary group are related by the inclusions:
U(n-1, 1) ⊃ U(n-1) ⊃ O(n-1)
U(n-1, 1) ⊃ SU(n-1, 1) ⊃ SU(n-1) ⊃ SO(n-1)
The respective real dimensions are:
U(n-1, 1) n²
U(n-1) (n-1)²
O(n-1) (n-1)(n-2)/2
SU(n-1) n(n-2)
SU(n-1) is a compact simple Lie group, that is connected and simply
connected. The maximal compact subgroup of U(n-1, 1) is
S(U(1) X U(n-1))
The Lie algebra of U(n-1, 1) is
u(n-1, 1) := { X ∈ gl(n, C): X† γ(n) + γ(n) X = 0 }
where gl(n, C) is the Lie algebra of GL(n, C).
These algebra elements X have the form:
| B |x> |
| |
|<x| iλ |
with λ ∈ R, |x> ∈ Hilb(n), B ∈ u(n-1), where u(n-1) is the Lie
algebra of the unitary group U(n-1).
The Lie algebra of U(G, n) is
u(G, n) := { X ∈ gl(n, C): X G(n) + G(n) X† = 0 }
These algebra elements X have the form:
| B |x>/√(n-1)|
| |
|√(n-1)<x| iλ |
with λ ∈ R, |x> ∈ Hilb(n), B ∈ u(n-1), where u(n-1) is the Lie
algebra of the group U(n-1). The algebra elements are right
skewhermitean with respect to G(n), or right G(n)-skew.
[Section XI]
The map of the elements of u(G, n)
X -> t-1(n) X t(n)
is a "conjugative map" from the algebra u(G, n) to the algebra u(n-1, 1).
The group U(G, n) and its coadjoint U(G-1, n) are subgroups of
GL(n, C), conjugate to each other and also conjugate to the subgroup
U(n-1, 1).
Complex Hyperbolic Spaces
Let M be a real hypersurface in Hilb(n) defined by
<x|γ(n)|x> = -1
U(n-1, 1) acts transitively on M, i. e., M is a homogeneous
space of U(n-1, 1)). That is:
∃ p ∈ M s.t. U(n-1, 1) p = M
It then follows this is true for every p in M; M is a single equivalence
class under transformations by U(n-1, 1). The subset Gp of M is called
an orbit of G in M, and homogeneous spaces have only one orbit.
If Hilb(n) were a real space, the condition
<x|γ(n)|x> = -1
would define two disconnected orbital hyperbolae, for the transitive
action of a Lorentz type pseudoorthogonal group SO(n-1, 1). For the
pseudounitary case, however, there is only one orbit.
Let
Gp = { g ∈ G: g|p> = |p> }
Gp is the set of elements of G that keep p invariant; it forms
a subgroup and is called the isotropy or stability group of G at p.
Let G be a topological group, H a closed subgroup of G.
Two elements g and g' are said to be congruent modulo G if the left
cosets gH and g'H coincide. This is an equivalence relation on G. The
space of equivalence classes is denoted by G/H.
[Chevalley 1946], [Pontryagin 1966].
If a group G acts transitively on M, then M can be reconstructed as being
isomorphic to the coset space of the isotropy group:
G/Gp -> M
A group G is said to act freely on a space M iff
g|z> = |z>
for some |z> ∈ M implies that g is the identity of G.
The group U(1) = { exp i θ } acts freely on M by
A -> A exp ( i θ ).
So the action of U(1) on M has no fixed points.
A group G is said to act effectively on a space M iff
every element of G that is not the identity element of the
abstract group does not act as the identity on M.
Let M' be the base manifold of the principle fibre bundle M
with group U(1).
[Kobayashi 1969], v. II p. 282,
[Steenrod 1951].
-------------------------------
|
|
M |
|
|
-------------------------------
π ↓ U(1)
M' --------------------.----------
For |z> ∈ M, the tangent space at z,
Tz(M) := { z ∈ M, w ∈ Hilb(n): Re( <w|γ(n)|z> ) = 0 }
So iz ∈ Tz(M). Define a subspace of Tz(M):
tz(M) := { z ∈ M, w ∈ Tz(M): Re( <w|γ(n)|iz> ) = 0 }
or equivalently
tz(M) := { z ∈ M, w ∈ Tz(M): <w|γ(n)|z> = 0 }
The restriction of <w|γ(n)|z> to tz(M)
is positive definite. There exists a connection in M so that
for z ∈ M, the Tz(M) are
the horizontal spaces of the bundle. The natural projection
π: M → M' induces a linear isomorphism of tz(M)
onto Tπ(z)(M'). The complex structures w → iw
on tz(M) with z ∈ M,
are compatible with the action of S1
and induce an almost complex structure J' on M' such that
π-1 i = J' π-1.
(π-1 is the inverse mapping of M' to M)
It turns out that J' is in fact integrable, so J' is a complex structure
and therefore that M' is a complex manifold. If we choose a negative
constant c, an inner product γ', in the tangent spaces
Tπ(z)(M')
of M' by
<Y|γ'(n)|X> := -(4/c) Re( <Y'|γ(n)|X'> )
with X' and Y' in tz(M), π-1(X') = X, π-1(Y') = Y,
the metric induced in the usual way
from the inner product is Hermitean with respect to J'.
See below for an explicit metric given in terms of the inhomogeneous
coordinates on M'.
The group U(n-1, 1) acts transitively on on M and therefore on M',
and leaves the structures γ'(n) and J' on M' invariant.
Let pn-1 = π-1(|n, n-1>).
The isotropy group of U(n-1, 1) at |n, n-1>
[Kobayashi 1969], v. II p. 283,
is a subgroup U(n-1) X U(1) of U(n-1, 1) defined by matrices of the form
|R 0 |
| |
|0 exp(iθ)|
where R ∈ U(n-1). There is an onto diffeomorphism
f : U(n-1, 1)/(U(n-1) X U(1)) -> M'
The action of U(n-1, 1) is not effective on M'. However, if
we take instead SU(n-1, 1), the subgroup of U(n-1, 1) whose
elements have determinant 1, it acts almost effectively.
[Actually, SU(n-1, 1)/Zⁿ acts effectively,
where Zⁿ is the
center of SU(n-1, 1) isomorphic to the cyclic group of
order n.]
For SU(n-1, 1), the isotropy group at |n, n-1> is
S(U(n) X U(1)) = {A ∈ U(n) X U(1): Det(A) = 1}
By taking conjugation by γ(n) as an involutive
automorphism, of SU(n-1, 1), then S(U(n) X U(1)) is the
subgroup of all elements left fixed by the conjugation.
So from the theory of symmetric spaces, M' is identified with
SU(n-1, 1)/S(U(n) X U(1)), is a symmetric space, and
is Hn-1(C), the complex hyperbolic
manifold of n-1 complex dimensions. It also happens to be
a Kaehlerian manifold with constant holomorphic sectional curvature c.
[Kobayashi 1969], v. II p. 285,
The construction of Hn-1(C)
is similar to that of the complex projective
space, Pn-1(C), where
Pn-1(C) = SU(n)/S(U(n-1) X U(1))
the space dual to Hn-1(C).
Then SU(n-1)/Zⁿ acts effectively on
Pn-1(C).
It may be of some interest in the present context to note that
Hn-1(C) may be identified with the
open unit ball Dn-1(C) in Hilb(n-1).
Dn-1(C) = { |ξ> ∈ Hilb(n-1): <ξ|ξ> < 1 }
by a mapping
p: <k|x> --> |x>/<n, n-1|x>
for k = 0, 1, ..., n-2. With |x> ∈ Hn-1(C).
The set
D = Dn-1(C)
is a bounded domain in the sense of
[Helgason 1962], p. 293.
It is convex, and there exists a function analytic in D
that cannot be analytically continued to the closure of D.
For an inspired tour de force on bounded complex domains, see
Tony Smith's page
In fact, the following definition of f(z)
f(z) = (1 - Σ α*k zk)-1
k
where (α1, α2, ..., αn-1) is a point on the boundary, is such a function. [Bochner 1948], p. 85. The space L²(D) of square integrable complex valued functions on D,
∫ |f|² dμ < ∞
D
where μ is the Lebesgue measure on R2n-2, is a Hilbert space with inner product given by
(f, g) = ∫ f(ζ) g*(ζ) dμ(ζ)
D
with ζ := (z1, z2, ..., zn-1),
with induced norm
‖f‖ = (f, f)1/2
The space H(D) of functions in L²(D) that are holomorphic in D is a closed linear subspace of L²(D) and therefore also forms a Hilbert space. If φk is any orthonormal basis of H(D), then the series defined by
∞
K(ζ, ξ*) := Σ φk(ζ) φk*(ξ*)
k
converges uniformly on every compact subset of D X D, is independent of the orthonormal basis chosen, and is "reproducing" in that for any f(ζ) ∈ H(D),
f(ζ) = ∫ K(ζ, ξ) f(ξ) dμ(ξ)
D
and K(ζ, ξ) is called the
Bergman kernel function for D
[Bergman 1933].
Define the complex tensor field,
h := Σ ∂k ∂j* log[K(ζ, ζ*)] dzk X dzj*
k,j
where dzk X dzj* is the complex covariant tensor field of rank 2 such that
(x, y) -> dzk(x) X dzj* (y)
where are complex covariant vector fields on D. Let g be the real part of the restriction of h to
T1(D) X T1(D)
where T1(D) is the space of real covariant vector fields on D. Then g is a Riemannian structure on D which is also Kaehlerian. Specifically in coordinate form,
gkj(ζ, ζ*) = 0,
gk*j*(ζ, ζ*) = 0,
gkj*(ζ, ζ*) = (1/2) ∂k ∂j* log K(ζ, ζ*)
A Kaehlerian manifold is a space of constant holomorphic sectional curvature c iff, for the nonzero curvature tensor components,
Kkjlm* = -(1/2) c (gkj* glm* + gkm* gjl*)
The bounded domain Dn(C) in Cn+1, admits a complete Kaehlerian metric of constant holomorphic sectional curvature c, where c is a negative constant, specified by the form of the differential line element,
ds² =
(1 - Σk wk wk*)(Σj dwj dwj*) - (Σk wk dwk*)(Σj dwj* dwj)
-(4/c) ----------------------------------------------
(1 - Σk wk wk*)²
and so the metric tensor components are given by
(1 - Σm wm wm*)δkj - w*k wj
gkj* = (2/c) -----------------------------
(1 - Σm wm wm*)²
where the (w0, w1, ..., wn-1) are coordinates for Cn.
For a Hermitean covariant metric of the form
φ δkj - w*k wj
gkj* = A ----------------
φ²
where A is some constant, the contravariant components are given by
(φ - S) δkj + wk w*j
gkj* = A-1 φ --------------------
(φ - S)
where
S := Σk w*k wk
and the global conditions
φ ≠ 0
φ ≠ S
hold. Since gkj*, and gkj* are inverses
Σ gkh* gjh* = δkj
h
[Kobayashi 1969], v. II p. 157,
For a Kaehlerian metric, the components of the associated affine
connection are given by
Γkji = gkh* (∂gh*j/∂wi)
and
Γk*j*i* = gk*h (∂ghj*/∂w*i)
The conventions of [Kobayashi 1969] are used.
If
φ = a(r² - S)
so that
∂φ/∂wi = -a w*i
the expression for the affine connection components becomes
Γkji =
a δkj w*i - δki w*j (a + 1)
= ------------------- - -------- wk w*j w*i
φ φ(φ - S)
and
(φ - S) = a r² - (a + 1) S
Passing to the boundary of Dn-1(C) is related to passing to a QM limit when n → ∞. The boundary of Dn-1(C) a compactification of Hn-1(C), is the unit sphere in Cn, while passing to the boundary in the limit as n → ∞ is from the viewpoint of Hn-1(C) is difficult to see since the limit of Hn-1(C) is the limit of the G(n)-null hyperspace, which is no longer null. The limiting procedure, however, becomes more manageable if considered after compactification. [Kobayashi 1969], II pp. 163, 169, 285, [Hermann 1970], [Bochner 1948], pp. 119-121, [Helgason 1962], p. 298, [Kobayashi 1972], p. 78.
The interior of the unit disk can also be mapped to a space of upper half planes, a Siegel domain by a fractional linear transformation.
Let
Sn-1(C) = {(w, v) ∈ Cn-1, : Im( v ) > |w|²}
with w ∈ Cn-2 and v ∈ C, be a Siegel Domain.
Then the fractional linear transformation
f(w, v) = (-iw(v - 1), -i(v + 1)/(v - 1))
maps Dn-1(C) onto Sn-1(C).
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