A Note on Iwasawa-Type Decomposition

Abstract

We study the Iwasawa-type decomposition of an open subset of SL(𝑛,ℂ) as SU(𝑝,𝑞)𝐴𝑁. We show that the dressing action of SU(𝑝,𝑞) is globally defined on the space of admissible elements in 𝐴𝑁. We also show that the space of admissible elements is a multiplicative subset of 𝐴𝑁. We establish a geometric criterion: the symmetrization of an admissible element maps the positive cone in ℂ^𝑛 into itself.

1. Introduction

In Poisson geometry, the groups SU(𝑝,𝑞) and 𝐴𝑁 (the upper-triangular subgroup of SL(𝑛,ℂ) with real positive diagonal entries) are naturally dual to each other [1]. Therefore, it is important to know the geometry of the orbits of the dressing action. We show that the right dressing action of SU(𝑝,𝑞) is globally defined on the open subset of the so-called admissible elements of 𝐴𝑁 (see Section 2).

We also show that the admissible elements can be characterized as follows: these are exactly those elements of 𝐴𝑁 whose symmetrization maps the closure of the positive cone in ℂ^𝑛 into the positive cone. In addition, we establish a useful fact that the set of admissible elements is a multiplicative subset of 𝐴𝑁.

2. Admissible Elements

Let 𝑝 and 𝑞 be positive integers and let 𝑛=𝑝+𝑞. Consider the space ℂ^𝑛 and the group 𝐺₀=SU(𝑝,𝑞), the group 𝐺₀ is the subgroup of 𝐺=SL(𝑛,ℂ), which preserves the following sesquilinear pairing on ℂ^𝑛: ⟨𝐱,𝐲⟩=^𝑝_𝑖=1𝑥_𝑖𝑦_𝑖−^𝑛_𝑗=𝑝+1𝑥_𝑗𝑦_𝑗. (2.1) Denote the corresponding norm by ‖𝐱‖. A vector 𝐱∈ℂ^𝑛 is said to be timelike if the square of its norm is positive, and spacelike, if it is negative.

Let 𝐴⊂𝐺 be the subgroup of positive real diagonal matrices and 𝑁 be the unipotent upper triangular subgroup. Denote by 𝔤, 𝔞, 𝔫, and 𝔤₀ the Lie algebras of 𝐺, 𝐴, 𝑁, and 𝐺₀. The Iwasawa-type decomposition for 𝐺 states that for an open dense subset 𝐺^∨ of 𝐺 one has [2, page 167]: 𝐺^∨=_{𝑤∈𝑊/𝑊0}𝐺₀𝑤̇𝐴𝑁, (2.2) where 𝑊=𝑊(𝔤,𝔞) is the Weyl group, 𝑊₀ is the subgroup of 𝑊 with representatives in 𝐾=SU(𝑛)∩𝐺₀, and 𝑤̇ is a representative of 𝑤∈𝑊/𝑊₀ in SU(𝑛).

An important question is to determine which elements of 𝐺 allow such a decomposition for a given choice of 𝑤. The case of particular interest is when 𝑤=1, since it is related to the dressing action in Poisson geometry.

Let 𝐽_𝑝,𝑞 be the diagonal matrix 𝐽_𝑝,𝑞=diag(1,1,…,1_𝑝,−1,−1,…,−1_𝑞). Introduce the following involution on the space of 𝑛×𝑛 matrices: 𝐴^†=𝐽_𝑝,𝑞𝐴^∗𝐽_𝑝,𝑞, (2.3) where 𝐴^∗ is the usual conjugate transpose. The Lie algebra 𝔤 splits, as a vector space, into the direct sum of ±1-eigenspaces of †: 𝔤=𝔮+𝔤₀. Let also 𝑄⊂𝐺 be the submanifold of elements satisfying 𝐴^†=𝐴. Clearly, exp(𝔮)⊂𝑄.

The next definition is quintessential for this paper. We say that 𝜆=diag𝜆₁,…,𝜆_𝑝,𝜇₁,…,𝜇_𝑞⊂𝔞 (2.4) is admissible if 𝜆_𝑖>𝜇_𝑗 for all 𝑖,𝑗. Clearly, using the action of 𝑊₀, one can arrange 𝜆_𝑖's and 𝜇_𝑗's in the nonincreasing order, and the condition of admissibility will become simply 𝜆_𝑝>𝜇₁. Same definition applies for 𝐴. Next, we say that an element 𝑋∈𝔮 is admissible, if it is 𝐺₀-conjugate to an admissible element in 𝔞. The set of admissible elements in 𝔮 form an open cone. Define the admissible elements in 𝑄 as the exponents of those in 𝔮. Finally, an element 𝑏∈𝐴𝑁 is called admissible, if it obtained from an admissible element of 𝐴 by the right dressing action.

The above definition stems from the work of Hilgert, Neeb and others, see, for example, [3]. Recall the definition of the right dressing action of 𝐺₀ on 𝐴𝑁. Let 𝑏∈𝐴𝑁 and 𝑔∈𝐺₀. Assume there exist 𝑏^′∈𝐴𝑁 and 𝑔^′∈𝐺₀ such that 𝑏𝑔=𝑔^′𝑏^′. In this case we write 𝑏^′=𝑏^𝑔.

One of the important properties of the set of admissible elements can be observed in the following result, which asserts that admissible elements map the timelike cone into itself.

Lemma 2.1. Let 𝑠∈𝑄 be admissible and let 𝐱∈ℂ^𝑛 be timelike. Then, 𝑠𝐱 is timelike as well.

Proof. Decompose 𝑠=𝑔⁻¹𝑒^𝜆𝑔, where 𝑔∈𝐺₀ and 𝜆=diag(𝜆₁,…,𝜆_𝑝,𝜇₁,…,𝜇_𝑞)⊂𝔞 is admissible, so that we have 𝜆_𝑖>𝜇₁≥𝜇_𝑗 for all 1≤𝑖≤𝑝 and 2≤𝑗≤𝑞.
Let 𝐲=𝑔𝐱. Since the group 𝐺₀ preserves the cone of timelike elements, 𝐲 is timelike as well and, denoting its coordinates 𝐲=𝑧₁,…,𝑧_𝑝,𝑤₁,…,𝑤_𝑞, (2.5) we see that ‖𝐲‖²=^𝑝_𝑖=1𝑧_𝑖²−^𝑞_𝑗=1𝑤_𝑗²=𝑟,𝑟∈ℝ,𝑟>0. (2.6)
Now, let us show that 𝑒^𝜆𝐲 is timelike, by using the above equation and expressing |𝑤₁|² in terms of the other coordinates: ^𝑝_𝑖=1𝑒^2𝜆𝑖𝑧_𝑖²−^𝑞_𝑗=1𝑒^2𝜇𝑗𝑤_𝑗²=^𝑝_𝑖=1𝑒^2𝜆𝑖−𝑒^2𝜇1𝑧_𝑖²+^𝑞_𝑗=2𝑒^2𝜇1−𝑒^2𝜇𝑗𝑤_𝑗²+𝑟𝑒^2𝜇1>0. (2.7) The last step in the proof is to notice that 𝑔⁻¹ preserves the timelike cone, and thus 𝑠𝐱=𝑔⁻¹𝑒^𝜆𝐲 is timelike.

Later on, in Proposition 4.4, we will show that conversely, admissible elements can be characterized by this property.

3. Gram-Schmidt Orthogonalization and Admissible Elements

Here we will give an explicit computational indication that if 𝜆∈𝔞 is admissible, then the whole orbit exp(𝜆)𝐺₀ admits a global decomposition, that is, exp(𝜆)𝐺₀⊂𝐺₀𝐴𝑁. Therefore, the right dressing action is globally defined on the set of admissible elements in 𝐴𝑁. Note that this is not true for the left dressing action, as a simple 2×2 example can show.

Next, we will use the Gram-Schmidt orthogonalization process to show that for any 𝑔∈𝐺₀ and admissible 𝜆, we have exp(𝜆)𝑔⊂𝐺₀𝐴𝑁. A short proof of this will be given in Section 4. Let us denote the columns of 𝑔 by 𝐰₁,…,𝐰_𝑛. The columns of 𝑔 are (pseudo-) orthonormal with respect to ⟨⋅,⋅⟩, and the first 𝑝 columns are timelike, and the last 𝑞 are spacelike.

Denote the columns of exp(𝜆)𝑔 by 𝐯₁,…,𝐯_𝑛. This element of 𝐺 is obtained from 𝑔 by multiplying the first row by 𝑒^𝜆1,…, the 𝑝th row by 𝑒^𝜆𝑝, the (𝑝+1)st row by 𝑒^𝜇1,…, and the last row by 𝑒^𝜇𝑞.

An important observation is that due to the admissibility of 𝜆, the first 𝑝 columns of exp(𝜆)𝑔 will remain timelike, however nothing definite can be said about the last 𝑞.

The decomposition exp(𝜆)𝑔=𝑠𝑏 with 𝑠∈𝐺₀ and 𝑏∈𝐴𝑁 is an analogue of the Gram-Schmidt orthogonalization process for the pseudometric ⟨⋅,⋅⟩.

Denote the columns of 𝑠 by 𝐮₁,…,𝐮_𝑛. Proving the existence of such decomposition amounts to showing that if we follow the Gram-Schmidt process, the first 𝑝 columns of 𝑠 will be timelike, and the last 𝑞 will be spacelike, and that the diagonal entries of 𝑏 will be positive real numbers. Let us denote the diagonal entries of 𝑏 by (𝑟₁,…,𝑟_𝑛) and the off-diagonal by 𝑚_𝑖𝑗, where 𝑚_𝑖𝑗=0 for 𝑖>𝑗.

Consider the first step of the Gram-Schmidt process, namely that 𝑟₁=‖𝐯₁‖ and 𝐮₁=𝐯₁/𝑟₁. Since 𝐯₁ is timelike, we see that 𝑟₁ is real positive and that 𝐮₁ is timelike.

Now we move to the second step, (obviously, under the assumption that 𝑝>1), which we consider in detail, because it lays the foundation for the other columns: 𝐯₂=𝑚₁₂𝐮₁+𝑟₂𝐮₂. (3.1) Here 𝑚₁₂=⟨𝐯₂,𝐮₁⟩ and 𝑟₂=‖𝐯₂−𝑚₁₂𝐮₁‖. In order to complete this step we need to show that the vector 𝐯₂−𝑚₁₂𝐮₁ is timelike.

Note that𝐯₂−𝑚₁₂𝐮₁²=𝐯₂²−𝐯₂,𝐮₁², (3.2) so we just need to show that this number is positive. Since ‖𝐮₁‖²=1, this is equivalent to showing that 𝐯₂²⋅𝐮₁²>𝐯₂,𝐮₁², (3.3) or, after multiplying both sides by 𝑟²₁, that𝐯₂²⋅𝐯₁²>𝐯₂,𝐯₁². (3.4)

Denote the coordinates of the vector 𝐰₁ by (𝑎₁,…,𝑎_𝑝,𝑏₁,…,𝑏_𝑞), and the coordinates of 𝐰₂ by (𝑐₁,…,𝑐_𝑝,𝑑₁,…,𝑑_𝑞). We have^𝑝_𝑖=1𝑎_𝑖²−^𝑞_𝑗=1𝑏_𝑗²=^𝑝_𝑖=1𝑐_𝑖²−^𝑞_𝑗=1𝑑_𝑗²=1,_𝑝𝑖=1𝑎_𝑖𝑐_𝑖−^𝑞_𝑗=1𝑏_𝑗𝑑_𝑗=0. (3.5)(3.6)

The coordinates of the vector 𝐯₁ are given by 𝑒^𝜆1𝑎₁,…,𝑒^𝜆𝑝𝑎_𝑝,𝑒^𝜇1𝑏₁,…,𝑒^𝜇𝑞𝑏_𝑞 (3.7) and 𝐯₂ by 𝑒^𝜆1𝑐₁,…,𝑒^𝜆𝑝𝑐_𝑝,𝑒^𝜇1𝑑₁,…,𝑒^𝜇𝑞𝑑_𝑞. (3.8)

The RHS of (3.4) now becomes as follows, using (3.5): ^𝑝_𝑖=1𝑒^2𝜆𝑖𝑎_𝑖²−^𝑞_𝑗=1𝑒^2𝜇𝑗𝑏_𝑗²⋅^𝑝_𝑖=1𝑒^2𝜆𝑖𝑐_𝑖²−^𝑞_𝑗=1𝑒^2𝜇𝑗𝑑_𝑗²=^𝑝_𝑖=1𝑒^2𝜆𝑖−𝑒^2𝜇1𝑎_𝑖²+^𝑞_𝑗=2𝑒^2𝜇1−𝑒^2𝜇𝑗𝑏_𝑗²+𝑒^2𝜇1⋅^𝑝_𝑖=1𝑒^2𝜆𝑖−𝑒^2𝜇1𝑐_𝑖²+^𝑞_𝑗=2𝑒^2𝜇1−𝑒^2𝜇𝑗𝑑_𝑗²+𝑒^2𝜇1, (3.9) which is strictly greater than ^𝑝_𝑖=1𝑒^2𝜆𝑖−𝑒^2𝜇1𝑎_𝑖²+^𝑞_𝑗=2𝑒^2𝜇1−𝑒^2𝜇𝑗𝑏_𝑗²⋅^𝑝_𝑖=1𝑒^2𝜆𝑖−𝑒^2𝜇1𝑐_𝑖²+^𝑞_𝑗=2𝑒^2𝜇1−𝑒^2𝜇𝑗𝑑_𝑗², (3.10) which in turn, by Cauchy-Schwarz, is greater or equal than ^𝑝_𝑖=1𝑒^2𝜆𝑖−𝑒^2𝜇1𝑎_𝑖𝑐_𝑖+^𝑞_𝑗=2𝑒^2𝜇1−𝑒^2𝜇𝑗𝑏_𝑗𝑑_𝑗², (3.11) which is exactly |⟨𝐯₁,𝐯₂⟩|², if we take into account (3.6).

Now, we will similarly show that 𝐮_𝑘 is timelike for any 𝑘≤𝑝. We have 𝐯_𝑘=^𝑘−1_ℓ=1𝑚_ℓ𝑘𝐮_ℓ+𝑟_𝑘𝐮_𝑘, (3.12) where 𝑚_ℓ𝑘=⟨𝐯_𝑘,𝐮_ℓ⟩.

This amounts to showing that 𝐯_𝑘²>^𝑘−1_ℓ=1𝐯_𝑘,𝐮_ℓ². (3.13)

Consider the subspace 𝑉⊂ℂ^𝑛 spanned by 𝐮₁,…,𝐮_𝑘−1, which is the same subspace that is spanned by 𝐯₁,…,𝐯_𝑘−1. It is isomorphic to the subspace 𝑊 spanned by 𝐰₁,…,𝐰_𝑘−1, and the explicit isomorphism is given by multiplying the first coordinate by 𝑒^−𝜆1,…, the 𝑝th by 𝑒^−𝜆𝑝, the (𝑝+1)st by 𝑒^−𝜇1,…, and the last by 𝑒^−𝜇𝑞.

Denote by 𝐲_𝑘=∑^𝑘−1_ℓ=1𝑚_ℓ𝑘𝐮_ℓ If ⟨𝐯_𝑘,𝐲_𝑘⟩=0, then by definition of 𝐲_𝑘 and 𝑚_ℓ𝑘 this would imply that 𝐯_𝑘 is orthogonal to all the 𝐮_𝑖 for 1≤𝑖≤𝑘−1, and thus all 𝑚_ℓ𝑘=0 and, similar to the case 𝑘=1, the vector 𝐮_𝑘=𝐯_𝑘/|𝐯_𝑘| is timelike as desired.

Thus, we can assume that 𝛼_𝑘=⟨𝐯_𝑘,𝐲_𝑘⟩=^𝑘−1_ℓ=1𝑚_ℓ𝑘²>0. (3.14)

Define 𝐲𝐮=^𝑘√𝛼_𝑘. (3.15) This is an essentially unique element of 𝑉 (up to a circle factor) with the property that: ⟨𝐯_𝑘,𝐮⟩⋅𝐮=^𝑘−1_ℓ=1𝑚_ℓ𝑘𝐮_ℓ. (3.16) Let 𝐰 also be the unit vector in 𝑊, which is the rescaled image of 𝐮 under the above homomorphism. The previous proof for the case 𝑘=2 now applies verbatim, with the role of 𝐰₁ played by 𝐰 and the role of 𝐯₂ played by 𝐯_𝑘.

Thus we have established that the first 𝑝 columns of 𝑠 are timelike vectors, and that 𝑟₁,…,𝑟_𝑝 are positive real numbers. Due to the Sylvester's law of inertia for quadratic forms, the only way to complete this to a (pseudo-) orthonormal basis with respect to ⟨⋅,⋅⟩ is to add 𝑞 spacelike vectors—therefore if we continue the Gram-Schmidt orthogonalization process, then the last 𝑞 columns of 𝑠 will be spacelike as required.

4. Iwasawa-Type Decomposition

Let us introduce more notation. Denote by 𝑄_adm and 𝔮_adm the sets of admissible elements in 𝑄 and 𝔮, respectively. Recall that 𝑄_adm=exp(𝔮_adm). The following result is straightforward.

Lemma 4.1. On these sets of admissible elements the map exp is a diffeomorphism.

Now, consider the symmetrization map Sym∶𝐴𝑁⟶𝑄,𝑌⟼𝑌^†𝑌. (4.1) This map sends the orbits of the dressing action to the conjugation orbits by the action of 𝐺₀ and therefore maps admissible elements to admissible. In fact, on the set of admissible elements, this map is, again, bijective, and, moreover, a simple computation of the differential can show that this is a diffeomorphism.

Let, as before, 𝜆∈𝔞 be admissible, let 𝑔₀∈𝐺₀, and let us find an explicit decomposition exp𝜆𝑔₀=𝑔^′₀𝑎𝑛, (4.2) with 𝑔^′₀∈𝐺₀, 𝑎∈𝐴, and 𝑛∈𝑁. Notice that the symmetrization map yields 𝑔₀⁻¹exp2𝜆𝑔₀=𝑛^†𝑎²𝑛. (4.3) The right hand side provides the Gauss decomposition (also known as the triangular or the LDV-factorization) of the left hand side. It exists if and only if the leading principal minors of the matrix on the left are nonzero. However, it was shown in [4, Proposition 4.1], that the eigenvalues of the principal minors of an admissible element satisfy certain interlacing conditions, similar to the Gelfand-Tsetlin conditions in the Hermitian symmetric case. In particular, since all the eigenvalues of 𝑠∈𝑄_adm are positive, then the eigenvalues of any leading principal minor would be positive as well, and therefore the Gauss decomposition exists.

The diagonal entries of 𝑎² are the ratios of the leading principal minors:𝑎²_𝑖𝑖=Δ_𝑖Δ_𝑖−1. (4.4)

Thus we also see that the entries of 𝑎² are positive, as required. Thus, we have established the following.

Proposition 4.2. If 𝜆∈𝔞 is admissible, then the whole orbit exp(𝜆)𝐺₀ admits a global decomposition, that is, exp(𝜆)𝐺₀⊂𝐺₀𝐴𝑁.

Another interesting property of (𝐴𝑁)_adm is that it is a multiplicative set.

Proposition 4.3. If two elements 𝑏₁ and 𝑏₂ from 𝐴𝑁 are admissible, 𝑏₁,𝑏₂∈(𝐴𝑁)_adm, then their product 𝑏₁𝑏₂ is admissible as well.

Proof. Applying the dressing action, we can actually assume that 𝑏₂=𝑎∈𝐴_adm. This follows from the fact that for 𝑔∈𝐺₀ and 𝑏₁,𝑏₂∈𝐴𝑁: 𝑔^𝑏1𝑏2=𝑔^𝑏2𝑏1,𝑏₁𝑏₂^𝑔=𝑏^(𝑔𝑏2)₁𝑏^𝑔₂. (4.5) Applying the symmetrization map to 𝑏₁𝑎, we obtain a new matrix 𝑓=𝑎𝑏^†₁𝑏₁𝑎∈𝑄, which we need to prove admissible.
Since the space 𝐴_adm is clearly connected and 𝑎 is admissible, consider a path 𝑎(𝑡) such that 𝑎(0)=I𝑑, 𝑎(1)=𝑎 and 𝑎(𝑡)∈𝐴_adm for 0<𝑡≤1. Also denote 𝑓(𝑡)=𝑎(𝑡)𝑏^†₁𝑏₁𝑎(𝑡).
Assume, on the contrary, that 𝑓(1)=𝑓 is not admissible and let ℰ⊂[0,1] be the (nonempty) subset defined by the property that 𝑓(𝑡) for 𝑡∈ℰ is not admissible. Then consider 𝜏=infℰ.
We recall [3] that the space 𝔮_adm forms a convex cone in 𝑞, therefore a simple infinitesimal computation can show that 𝑄_adm is a connected open subset of 𝑄 and any element from the boundary of 𝑄_adm, such as 𝑓(𝜏), has the property that its eigenvalues remain real and, moreover, the lowest eigenvalue 𝜆_𝑝 corresponding to the timelike cone and the highest eigenvalue 𝜇₁ corresponding to the spacelike cone collide: 𝜆_𝑝=𝜇₁=𝛽. Thus, there exists a whole plane of eigenvectors containing vectors from both ℂ^𝑛₊ and ℂ^𝑛₋. Thus it must contain an eigenvector, which we denote by 𝐳, from the null cone.
For this eigenvector we have the following equation: 𝑓(𝜏)𝐳=𝛽𝐳. Or, equivalently, 𝑎(𝜏)𝑏^†₁𝑏₁𝑎(𝜏)𝐳=𝛽𝐳. (4.6) Note that both 𝑎(𝜏) and 𝑏^†₁𝑏₁, being admissible, map the timelike cone ℂ^𝑛₊ into itself by Lemma 2.1. Similar proof shows that they map the null cone minus the origin ℂ^𝑛₀⧵{0} inside the timelike cone. Therefore, the element 𝑎(𝜏)(𝑏^†₁𝑏₁)𝑎(𝜏) also maps ℂ^𝑛₀⧵{0} inside of ℂ^𝑛₊ and thus 𝐳 cannot be its eigenvector.

In geometric terms, we have established the following characterization of the space of admissible elements 𝑄_adm.

Proposition 4.4. An element 𝑠∈𝑄 is admissible if and only if it has real positive eigenvalues and maps the closure of the timelike cone ℂ^𝑛₊=ℂ^𝑛₊∪ℂ^𝑛₀ into the timelike cone ℂ^𝑛₊ (plus the origin).

Notice that since 𝑎 from the above proof is admissible diagonal, the pseudohermitian analogue of the Rayleigh-Ritz ratio ℛ_𝐴=𝐱^†𝐴𝐱𝐱^†𝐱 (4.7) for the matrices 𝑓 and 𝑠=𝑏^†₁𝑏₁ will have the following properties. Let 𝐱∈ℂ^𝑛₊ be timelike, then, as we saw earlier, 𝑎𝐱 is timelike as well with a bigger norm by Lemma 2.1.

Thus, for 𝐱∈ℂ^𝑛₊, we have ℛ_𝑎𝑠𝑎(𝐱)=𝐱^†𝑎𝑠𝑎𝐱𝐱^†𝐱>(𝑎𝐱)^†𝑠(𝑎𝐱)^†(𝑎𝐱)^†𝑎𝐱=ℛ_𝑠(𝑎𝐱). (4.8) It follows since 𝑎 maps ℂ^𝑛₊ to itself, that min_{𝐱∈ℂ𝑛+}ℛ_𝑠(𝐱)≤min_{𝐱∈ℂ𝑛+}ℛ_𝑠(𝑎𝐱)<min_{𝐱∈ℂ𝑛+}ℛ_𝑎𝑠𝑎(𝐱). (4.9) From [5, Theorem 2.1], it follows that 𝜆₁(𝑎𝑠𝑎)>𝜆₁(𝑠). Similar inequalities can be established for other eigenvalues.

The notion of admissibility can be extended to the whole group 𝐺=SL(𝑛,ℂ). We say that 𝑔∈𝐺 is admissible, if it decomposes as 𝑔=ℎ𝑏, with ℎ∈𝐺₀ and an admissible 𝑏∈(𝐴𝑁)_adm. Such a decomposition, if exists, is clearly unique. Note, moreover, that 𝑔^†𝑔=𝑏^†𝑏. The singular admissible spectrum of 𝑔 are the square roots of the spectrum of 𝑔^†𝑔. (Note of warning: one should not define 𝑄_adm by inducing this definition from 𝐺, but rather as we did previously.)

Suppose that 𝑔₁, 𝑔₂ are two admissible elements from 𝐺 such that 𝑔₁=ℎ₁𝑏₁ and 𝑔₂=ℎ₂𝑏₂. Then we have 𝑔₁𝑔₂=ℎ₁𝑏₁ℎ₂𝑏₂=ℎ₁ℎ^𝑏1₂𝑏^ℎ2₁𝑏₂, where the powers mean the dressing actions. This is possible, because 𝑏₁ is admissible. Thus we have established that the product 𝑔₁𝑔₂ is admissible if and only if the product 𝑏^ℎ2₁𝑏₂ is such.

5. Example of the Group SU(1,1)

The Poisson geometry related to the group SU(1,1) was considered in detail in [6]. Here we just recall several facts to illustrate the results of this paper. First of all, the space 𝔮_adm is the following convex cone of matrices 𝑧𝑥+𝑖𝑦−𝑥+𝑖𝑦−𝑧∶𝑥,𝑦,𝑧∈ℝ,𝑧²−𝑥²−𝑦²>0,𝑧>0. (5.1) Next, we define 𝑄_adm=exp(𝔮_adm). Consider an element 𝑡₁𝑚−𝑚𝑡₂∈𝑄, (5.2) where 𝑡₁,𝑡₂∈ℝ, and 𝑚∈ℂ satisfy the determinant condition 𝑡₁𝑡₂+|𝑚|²=1. It is admissible if and only if its eigenvalues are real and positive, and, moreover, the eigenvalue for the timelike cone is greater than the other one. This translates into the following two conditions on the coefficients of this matrix: 𝑡₁+𝑡₂>2,𝑡₁>1. (5.3) Next, an element 𝑟𝑛0𝑟⁻¹∈𝐴𝑁 (5.4) is admissible if and only if 𝑟>1,𝑟²+𝑟⁻²−|𝑛|²>2. (5.5)

Now, let us write down the Iwasawa-type decomposition 𝐺⊃𝐺₀𝐴𝑁 explicitly. Let us consider a general element of SL(2,ℂ): 𝑎𝑏𝑐𝑑∶𝑎,𝑏,𝑐,𝑑∈ℂ,𝑎𝑑−𝑏𝑐=1. (5.6) The condition that this general element admits such a decomposition is simply |𝑎|>|𝑐|: =𝑎𝑎𝑏𝑐𝑑|𝑎|²−|𝑐|²𝑐|𝑎|²−|𝑐|²𝑐|𝑎|²−|𝑐|²𝑎|𝑎|²−|𝑐|²⋅|𝑎|²−|𝑐|²𝑏−𝑐|𝑎|²−|𝑐|²|𝑎|²−|𝑐|²𝑎01|𝑎|²−|𝑐|². (5.7) Thus, it is quite easy to see that for any element 𝑔∈𝐺 and exp(𝜆)∈𝐴_adm, the element exp(𝜆)𝑔∈𝐺₀𝐴𝑁. Explicitly, if 𝑔=𝑢𝑣𝑣𝑢𝜆=𝑒,exp^𝜆00𝑒^−𝜆,𝜆>0, (5.8) then 𝜆𝑒exp𝑔=^𝜆𝑢𝑒^𝜆𝑣𝑒^−𝜆𝑣𝑒^−𝜆𝑢 (5.9) is an admissible element of 𝐺.

The statemement that the admissibility of two elements 𝑏₁ and 𝑏₂ of 𝐴𝑁⊂SL(2,ℂ) implies the admissibility of their product is also a short computational affair.