According to the "elementary approach" to bounded symmetric space of Koecher, Hermitian symmetric spaces of noncompact type can be realized in the complexification of a Euclidean Jordan algebra
E as either the open unit ball for the spectral norm, a bounded domain, or as the open tube domain , where
C is the positive open cone in
E. In the simplest case where
E =
R, the complexification of
E is just
C, the bounded domain corresponds to the open unit disk and the tube domain to the upper half plane. Both these spaces have transitive groups of biholomorphisms given by Möbius transformations, corresponding to matrices in or . They both lie in the Riemann sphere {{math|1=
C ∪ {∞}}}, the standard one-point compactification of
C. Moreover, the symmetry groups are all particular cases of Möbius transformations corresponding to matrices in . This complex Lie group and its maximal compact subgroup act transitively on the Riemann sphere. The groups are also algebraic. They have distinguished generating subgroups and have an explicit description in terms of generators and relations. Moreover, the Cayley transform gives an explicit Möbius transformation from the open disk onto the upper half plane. All these features generalize to arbitrary Euclidean Jordan algebras. The compactification and complex Lie group are described in the next section and correspond to the dual Hermitian symmetric space of compact type. In this section only the symmetries of and between the bounded domain and tube domain are described. Jordan frames provide one of the main Jordan algebraic techniques to describe the symmetry groups. Each Jordan frame gives rise to a product of copies of
R and
C. The symmetry groups of the corresponding open domains and the compactification—polydisks and polyspheres—can be deduced from the case of the unit disk, the upper halfplane and Riemann sphere. All these symmetries extend to the larger Jordan algebra and its compactification. The analysis can also be reduced to this case because all points in the complex algebra (or its compactification) lie in an image of the polydisk (or polysphere) under the unitary structure group.
Definitions Let be a Euclidean Jordan algebra with complexification . The unit ball or disk
D in is just the convex bounded open set of elements such the ||
a|| \displaystyle{g=\begin{pmatrix}\alpha & \beta \\ \gamma & \delta\end{pmatrix},} then :\displaystyle{g(z)=(\alpha z +\beta)(\gamma z +\delta)^{-1}.} Similarly the group SL(2,
R) acts by Möbius transformations on the circle
R ∪ {∞}, the one-point compactification of
R. Let
k =
R or
C. Then SL(2,
k) is generated by the three subgroups of lower and upper unitriangular matrices,
L and
U', and the diagonal matrices
D. It is also generated by the lower (or upper) unitriangular matrices, the diagonal matrices and the matrix :\displaystyle{J=\begin{pmatrix}0 & 1 \\ -1 & 0\end{pmatrix}.} The matrix
J corresponds to the Möbius transformation and can be written :\displaystyle{J=\begin{pmatrix}1 & 0 \\ -1 & 1\end{pmatrix}\begin{pmatrix}1 & 1 \\ 0 & 1\end{pmatrix} \begin{pmatrix}1 & 0 \\ -1 & 1\end{pmatrix}.} The Möbius transformations fixing ∞ are just the upper triangular matrices
B =
UD =
DU. If
g does not fix ∞, it sends ∞ to a finite point
a. But then
g can be composed with an upper unitriangular matrix to send
a to 0 and then with
J to send 0 to infinity. This argument gives one of the simplest examples of the
Bruhat decomposition: :\displaystyle{\mathbf{SL}(2,k) = \mathbf{B} \cup \mathbf{B}\cdot J\cdot \mathbf{B},} the double coset decomposition of . In fact the union is disjoint and can be written more precisely as :\displaystyle{\mathbf{SL}(2,k) = \mathbf{B} \cup \mathbf{B} \cdot J\cdot\mathbf{U},} where the product occurring in the second term is direct. Now let :\displaystyle{T(\beta)=\begin{pmatrix}1 & \beta \\ 0 & 1\end{pmatrix}.} Then :\displaystyle{\begin{pmatrix}\alpha & 0 \\ 0 & \alpha^{-1}\end{pmatrix} = JT(\alpha^{-1}) JT(\alpha)JT(\alpha^{-1}).} It follows is generated by the group of operators and
J subject to the following relations: • is an additive homomorphism • is a multiplicative homomorphism • • • The last relation follows from the definition of . The generator and relations above is fact gives a presentation of . Indeed, consider the free group Φ generated by
J and with
J of order 4 and its square central. This consists of all products for . There is a natural homomorphism of Φ onto . Its kernel contain the normal subgroup Δ generated by the relations above. So there is a natural homomorphism of Φ/Δ onto . To show that it is injective it suffices to show that the Bruhat decomposition also holds in . It is enough to prove the first version, since the more precise version follows from the commutation relations between
J and . The set is invariant under inversion, contains operators and
J, so it is enough to show it is invariant under multiplication. By construction it is invariant under multiplication by
B. It is invariant under multiplication by
J because of the defining equation for . In particular the center of consists of the scalar matrices and it is the only non-trivial normal subgroup of , so that {{math|1=PSL(2,
k) = SL(2,
k)/{±
I}}} is
simple. In fact if is a normal subgroup, then the Bruhat decomposition implies that is a maximal subgroup, so that either is contained in or . In the first case fixes one point and hence every point of {{math|1=
k ∪ {∞}}}, so lies in the center. In the second case, the
commutator subgroup of is the whole group, since it is the group generated by lower and upper unitriangular matrices and the fourth relation shows that all such matrices are commutators since . Writing with in and in , it follows that . Since and generate the whole group, . But then . The right hand side here is Abelian while the left hand side is its own commutator subgroup. Hence this must be the trivial group and . Given an element
a in the complex Jordan algebra , the unital Jordan subalgebra is associative and commutative. Multiplication by
a defines an operator on which has a spectrum, namely its set of complex eigenvalues. If is a complex polynomial, then is defined in . It is invertible in if and only if it is invertible in , which happen precisely when does not vanish on the spectrum of . This permits
rational functions of to be defined whenever the function is defined on the spectrum of . If and are rational functions with and defined on , then is defined on and . This applies in particular to complex Möbius transformations which can be defined by . They leave invariant and, when defined, the group composition law holds. (In the next section complex Möbius transformations will be defined on the compactification of .) Given a primitive idempotent in with Peirce decomposition :\displaystyle{E=E_1(e)\oplus E_{1/2}(e)\oplus E_0(e),\,\,\,\, A=A_1(e)\oplus A_{1/2}(e)\oplus A_0(e).} the action of by Möbius transformations on can be extended to an action on
A so that the action leaves invariant the components and in particular acts trivially on . If is the projection onto , the action is given be the formula :\displaystyle{g(z e\oplus x_{1/2} \oplus x_0)={\alpha z +\beta \over \gamma z +\delta}\cdot e \oplus (\gamma z +\delta)^{-1} x_{1/2} \oplus x_0 - (\gamma z + \delta)^{-1}P_0(x_{1/2}^2).} For a Jordan frame of primitive idempotents , the actions of associated with different commute, thus giving an action of . The diagonal copy of gives again the action by Möbius transformations on .
Cayley transform The Möbius transformation defined by :\displaystyle{C(z)=i{1+z\over 1-z}=-i +{2i\over 1 - z}} is called the
Cayley transform. Its inverse is given by :\displaystyle{P(w)={w-i\over w+i} = 1 -{2i\over w+i}.} The inverse Cayley transform carries the real line onto the circle with the point 1 omitted. It carries the upper halfplane onto the unit disk and the lower halfplane onto the complement of the closed unit disk. In
operator theory the mapping takes self-adjoint operators
T onto unitary operators
U not containing 1 in their spectrum. For matrices this follows because unitary and self-adjoint matrices can be diagonalized and their eigenvalues lie on the unit circle or real line. In this finite-dimensional setting the Cayley transform and its inverse establish a bijection between the matrices of operator norm less than one and operators with imaginary part a positive operator. This is the special case for of the Jordan algebraic result, explained below, which asserts that the Cayley transform and its inverse establish a bijection between the bounded domain and the tube domain . In the case of matrices, the bijection follows from resolvent formulas. In fact if the imaginary part of is positive, then is invertible since :\displaystyle{\|(T+iI)x\|^2= \|(T-iI)x\|^2 + 4(\mathrm{Im}(T)x,x).} In particular, setting , :\displaystyle{\|y\|^2= \|P(T)y\|^2 + 4(\mathrm{Im}(T)x,x).} Equivalently :\displaystyle{I-P(T)^*P(T) = 4 (T^* -iI)^{-1}[\mathrm{Im}\,T] (T+iI)^{-1}} is a positive operator, so that ||
P(
T)|| \displaystyle{\mathrm{Im}\,C(U)=(2i)^{-1}[C(U)-C(U)^*] = (1-U^*)^{-1}[I -U^*U](I-U)^{-1}.} Since the Cayley transform and its inverse commute with the transpose, they also establish a bijection for symmetric matrices. This corresponds to the Jordan algebra of symmetric complex matrices, the complexification of . In the above resolvent identities take the following form: :\displaystyle{Q(1-u^*)Q(C(u)+C(u^*))Q(1-u)=-4B(u^*,u)} and equivalently :\displaystyle{ 4Q(\mathrm{Im} \,a) = Q(a^*-i)B(P(a)^*,P(a))Q(a+i),} where the Bergman operator is defined by with . The inverses here are well defined. In fact in one direction is invertible for ||
u|| \displaystyle{Q(a)Q(a^{-1}+b^{-1})Q(b)=Q(a+b).} In fact in this case the
relations for a quadratic Jordan algebra imply :\displaystyle{R(a,b)=2Q(a)Q(a^{-1},b)=2Q(a,b^{-1})Q(b)} so that :\displaystyle{B(a,b)=Q(a)Q(a^{-1}-b)=Q(b^{-1} -a)Q(b).} The equality of the last two terms implies the identity, replacing by . Now set and . The resolvent identity is a special case of the more following more general identity: :\displaystyle{Q(1-x)Q(C(x)+C(y))Q(1-y)=-4B(x,y).} In fact :\displaystyle{C(x)+C(y)=-2i(1-a^{-1} -b^{-1}),} so the identity is equivalent to :\displaystyle{Q(a)Q(1-a^{-1} -b^{-1})Q(b)=B(1-a,1-b).} Using the identity above together with , the left hand side equals . The right hand side equals . These are equal because of the formula .
Automorphism group of bounded domain If lies in the bounded domain , then is invertible. Since is invariant under multiplication by scalars of modulus ≤ 1, it follows that is invertible for |λ| ≥ 1. Hence for ||
a|| ≤ 1, is invertible for |λ| > 1. It follows that the Möbius transformation is defined for ||
a|| ≤ 1 and in . Where defined it is injective. It is holomorphic on . By the
maximum modulus principle, to show that maps onto it suffices to show it maps onto itself. For in that case and its inverse preserve so must be surjective. If with in , then lies in . This is a commutative associative algebra and the spectral norm is the supremum norm. Since with |ς
i| = 1, it follows that where |
g(ς
i)| = 1. So lies in . This is a direct consequence of the definition of the spectral norm. This is already known for the Möbius transformations, i.e. the diagonal in . It follows for diagonal matrices in a fixed component in because they correspond to transformations in the unitary structure group. Conjugating by a Möbius transformation is equivalent to conjugation by a matrix in that component. Since the only non-trivial normal subgroup of is its center, every matrix in a fixed component carries onto itself. Given an element in an transformation in the identity component of the unitary structure group carries it in an element in with supremum norm less than 1. An transformation in the carries it onto zero. Thus there is a transitive group of biholomorphic transformations of . The symmetry is a biholomorphic Möbius transformation fixing only 0. If is a biholomorphic self-mapping of with and derivative at 0, then must be the identity. If not, has Taylor series expansion with homogeneous of degree and . But then . Let be a functional in of norm one. Then for fixed in , the holomorphic functions of a complex variable given by must have modulus less than 1 for |
w|
D is the set of all points with . The orbit of these points under the unitary structure group is the whole of . The Cartan decomposition follows because is the stabilizer of 0 in . In fact the only point fixed by (the identity component of)
KD in
D is 0. Uniqueness implies that the
center of
GD must fix 0. It follows that the center of
GD lies in
KD. The center of
KD is isomorphic to the circle group: a rotation through θ corresponds to multiplication by
eiθ on
D so lies in {{math|SU(1,1)/{±1}}}. Since this group has trivial center, the center of
GD is trivial. In fact any larger compact subgroup would intersect
AD non-trivially and it has no non-trivial compact subgroups. Note that
GD is a Lie group (see below), so that the above three statements hold with
GD and
KD replaced by their identity components, i.e. the subgroups generated by their one-parameter cubgroups. Uniqueness of the maximal compact subgroup up to conjugacy follows from
a general argument or can be deduced for classical domains directly using
Sylvester's law of inertia following . For the example of Hermitian matrices over
C, this reduces to proving that is up to conjugacy the unique maximal compact subgroup in . In fact if , then is the subgroup of preserving
W. The restriction of the hermitian form given by the inner product on minus the inner product on . On the other hand, if is a compact subgroup of , there is a -invariant inner product on obtained by averaging any inner product with respect to Haar measure on . The Hermitian form corresponds to an orthogonal decomposition into two subspaces of dimension both invariant under with the form positive definite on one and negative definite on the other. By Sylvester's law of inertia, given two subspaces of dimension on which the Hermitian form is positive definite, one is carried onto the other by an element of . Hence there is an element of such that the positive definite subspace is given by . So leaves invariant and . A similar argument, with
quaternions replacing the complex numbers, shows uniqueness for the symplectic group, which corresponds to Hermitian matrices over
R. This can also be seen more directly by using
complex structures. A complex structure is an invertible operator
J with
J2 = −
I preserving the symplectic form
B and such that −
B(
Jx,
y) is a real inner product. The symplectic group acts transitively on complex structures by conjugation. Moreover, the subgroup commuting with
J is naturally identified with the unitary group for the corresponding complex inner product space. Uniqueness follows by showing that any compact subgroup
K commutes with some complex structure
J. In fact, averaging over Haar measure, there is a
K-invariant inner product on the underlying space. The symplectic form yields an invertible skew-adjoint operator
T commuting with
K. The operator
S = −
T2 is positive, so has a unique positive square root, which commutes with
K. So
J =
S−1/2
T, the phase of
T, has square −
I and commutes with
K.
Automorphism group of tube domain There is a
Cartan decomposition for
GT corresponding to the action on the tube
T =
E +
iC: :\displaystyle{G_T=K_T A_T K_T.} •
KT is the stabilizer of
i in
iC ⊂
T, so a maximal compact subgroup of
GT. Under the Cayley transform,
KT corresponds to
KD, the stabilizer of 0 in the bounded symmetric domain, where it acts linearly. Since
GT is semisimple, every
maximal compact subgroup is conjugate to
KT. • The center of
GT or
GD is trivial. In fact the only point fixed by
KD in
D is 0. Uniqueness implies that the
center of
GD must fix 0. It follows that the center of
GD lies in
KD and hence that the center of
GT lies in
KT. The center of
KD is isomorphic to the circle group: a rotation through θ corresponds to multiplication by
eiθ on
D. In Cayley transform it corresponds to the
Möbius transformation z ↦ (
cz +
s)(−
sz +
c)−1 where
c = cos θ/2 and
s = sin θ/2. (In particular, when θ = π, this gives the symmetry
j(
z) = −
z−1.) In fact all Möbius transformations
z ↦ (α
z + β)(−γ
z + δ)−1 with αδ − βγ = 1 lie in
GT. Since PSL(2,
R) has trivial center, the center of
GT is trivial. •
AT is given by the linear operators
Q(
a) with
a = Σ α
i ei with α
i > 0. In fact the Cartan decomposition for follows from the decomposition for . Given in , there is an element in , the identity component of , such that with . Since ||
z||
T corresponding to the action on the tube
T =
E +
iC: :\displaystyle{G_T=K_T A_T N_T.} •
KT is the stabilizer of
i in
iC ⊂
T. •
AT is given by the linear operators
Q(
a) where
a = Σ α
i ei with α
i > 0. •
NT is a lower unitriangular group on
EC. It is the semidirect product of the unipotent triangular group
N appearing in the Iwasawa decomposition of
G (the symmetry group of
C) and
N0 =
E, group of translations
x ↦
x +
b. The group
S =
AN acts on
E linearly and conjugation on
N0 reproduces this action. Since the group
S acts simply transitively on
C, it follows that
ANT=
S⋅
N0 acts simply transitively on
T =
E +
iC. Let
HT be the group of
biholomorphisms of the tube
T. The Cayley transform shows that is isomorphic to the group
HD of biholomorphisms of the bounded domain
D. Since
ANT acts simply transitively on the tube
T while
KT fixes
ic, they have trivial intersection. Given
g in
HT, take
s in
ANT such that
g−1(
i)=
s−1(
i). then
gs−1 fixes
i and therefore lies in
KT. Hence
HT =
KT ⋅
A⋅
NT. So the product is a group.
Lie group structure By a result of
Henri Cartan,
HD is a Lie group. Cartan's original proof is presented in . It can also be deduced from the fact the
D is complete for the
Bergman metric, for which the isometries form a Lie group; by
Montel's theorem, the group of biholomorphisms is a closed subgroup. That
HT is a Lie group can be seen directly in this case. In fact there is a finite-dimensional 3-graded Lie algebra \mathfrak{g}_T of vector fields with an involution σ. The Killing form is negative definite on the +1 eigenspace of σ and positive definite on the −1 eigenspace. As a group
HT normalizes \mathfrak{g}_T since the two subgroups
KT and
ANT do. The +1 eigenspace corresponds to the Lie algebra of
KT. Similarly the Lie algebras of the linear group
AN and the affine group
N0 lie in \mathfrak{g}_T. Since the group
GT has trivial center, the map into GL(\mathfrak{g}_T) is injective. Since
KT is compact, its image in GL(\mathfrak{g}_T) is compact. Since the Lie algebra \mathfrak{g}_T is compatible with that of
ANT, the image of
ANT is closed. Hence the image of the product is closed, since the image of
KT is compact. Since it is a closed subgroup, it follows that
HT is a Lie group. ==Generalizations==