A Note on Iwasawa Type Decomposition

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International Journal of Mathematics and Mathematical Sciences Volume 2011, Article ID 135167,10pages

doi:10.1155/2011/135167

Research Article

A Note on Iwasawa-Type Decomposition

Philip Foth

1, 2

1_{CEGEP, Champlain St. Lawrence, QC, Canada G1V 4K2}

2_{Department of Mathematics, The University of Arizona, Tucson, AZ 85721-0089, USA}

Correspondence should be addressed to Philip Foth,[email protected]

Received 2 December 2010; Accepted 25 February 2011

Academic Editor: Hans Engler

Copyrightq2011 Philip Foth. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We study the Iwasawa-type decomposition of an open subset of SLn, as SUp, qAN. We show that the dressing action of SUp, qis globally defined on the space of admissible elements inAN. We also show that the space of admissible elements is a multiplicative subset ofAN. We establish a geometric criterion: the symmetrization of an admissible element maps the positive cone in n

into itself.

1. Introduction

In Poisson geometry, the groups SUp, qandANthe upper-triangular subgroup of SLn,

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 SUp, qis globally defined on the open subset of the so-called admissible

elements ofANseeSection 2.

We also show that the admissible elements can be characterized as follows: these are

exactly those elements ofANwhose symmetrization maps the closure of the positive cone

in n _{into the positive cone. In addition, we establish a useful fact that the set of admissible}

elements is a multiplicative subset ofAN.

2. Admissible Elements

Let p and qbe positive integers and let n pq. Consider the space n _{and the group}

G0 SUp, q, the groupG0is the subgroup ofG SLn, , which preserves the following

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x,yp i1

xiy_i−

n

jp1

xjy_j. 2.1

Denote the corresponding norm byx. A vectorx∈ n is said to be timelike if the square of

its norm is positive, and spacelike, if it is negative.

LetA⊂Gbe the subgroup of positive real diagonal matrices andNbe the unipotent

upper triangular subgroup. Denote by ,,, and 0the Lie algebras ofG,A,N, andG0. The

Iwasawa-type decomposition forGstates that for an open dense subsetG∨ofGone has2,

page 167:

G∨

w∈W/W0

G0wAN,˙ _2.2

where W W , is the Weyl group, W0 is the subgroup of W with representatives in

KSUn∩G0, and ˙wis a representative ofw∈W/W0in SUn.

An important question is to determine which elements ofGallow such a

decomposi-tion for a given choice ofw. The case of particular interest is whenw1, since it is related to

the dressing action in Poisson geometry.

Let Jp,q be the diagonal matrix Jp,q diag1,1, . . . ,1

p

,−1, −1, . . . , −1

q

. Introduce the

following involution on the space ofn×nmatrices:

A†Jp,qA∗Jp,q, 2.3

whereA∗is the usual conjugate transpose. The Lie algebra splits, as a vector space, into the

direct sum of±1-eigenspaces of†: 0. Let alsoQ⊂Gbe the submanifold of elements

satisfyingA†A. Clearly, exp⊂Q.

The next definition is quintessential for this paper. We say that

λdiagλ1, . . . , λp, μ1, . . . , μq ⊂ 2.4

is admissible ifλi> μjfor alli, j. Clearly, using the action ofW0, one can arrangeλi’s andμj’s in

the nonincreasing order, and the condition of admissibility will become simplyλp> μ1. Same

definition applies forA. Next, we say that an elementX∈is admissible, if it isG0-conjugate

to an admissible element in. The set of admissible elements inform an open cone. Define

the admissible elements inQas the exponents of those in. Finally, an elementb ∈AN is

called admissible, if it obtained from an admissible element ofAby 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 onAN. Let b ∈ AN

andg∈G0. Assume there existb∈ANandg∈G0such thatbggb. In this case we write

bbg.

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.

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Proof. Decompose s g−1eλ_{g, where} _g _∈ _G0 _and _λ _{diagλ1, . . . , λp, μ1, . . . , μq} _⊂

is

admissible, so that we haveλi> μ1≥μjfor all 1≤i≤pand 2≤j≤q.

Letygx. Since the groupG0preserves the cone of timelike elements,y is timelike as

well and, denoting its coordinates

yz1, . . . , zp, w1, . . . , wq , 2.5

we see that

y2p

i1

|zi|2₋q

j1

_wj2

r, r∈, r >0. 2.6

Now, let us show thateλ_{y is timelike, by using the above equation and expressing}_|w_1|2

in terms of the other coordinates:

p

i1

e2λi|zi|2−

q

j1

e2μj_wj2

p

i1

e2λi−_e2μ1

|zi|2q

j2

e2μ1−_e2μj_wj2_re2μ1 _>_0. 2.7

The last step in the proof is to notice thatg−1preserves the timelike cone, and thussxg−1eλ_y

is timelike.

Later on, inProposition 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 ifλ ∈ is admissible, then the

whole orbit expλG0admits a global decomposition, that is, expλG0 ⊂G0AN. Therefore,

the right dressing action is globally defined on the set of admissible elements inAN. 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 expλg ⊂ G0AN. A short proof of this will be given

inSection 4. Let us denote the columns ofgbyw1, . . . ,wn. The columns ofg are

pseudo-orthonormal with respect to ·,·, and the first p columns are timelike, and the last qare

spacelike.

Denote the columns of expλgbyv1, . . . ,vn. This element ofGis obtained fromgby

multiplying the first row byeλ1_{, . . ., the}_{pth row by}_eλp_{, the}p1st row by_eμ1_{, . . ., and the}

last row byeμq_.

An important observation is that due to the admissibility ofλ, the firstpcolumns of

expλgwill remain timelike, however nothing definite can be said about the lastq.

The decomposition expλg sbwith s ∈ G0 and b ∈ AN is an analogue of the

Gram-Schmidt orthogonalization process for the pseudometric·,·.

Denote the columns ofs byu1, . . . ,un. Proving the existence of such decomposition

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be timelike, and the lastqwill be spacelike, and that the diagonal entries ofbwill be positive

real numbers. Let us denote the diagonal entries ofbbyr1, . . . , rnand the oﬀ-diagonal by

mij, wheremij 0 fori > j.

Consider the first step of the Gram-Schmidt process, namely thatr1 v1andu1

v1/r1. Sincev1is timelike, we see thatr1is real positive and thatu1is timelike.

Now we move to the second step,obviously, under the assumption thatp >1, which

we consider in detail, because it lays the foundation for the other columns:

v2 m12u1r2u2. 3.1

Herem12 v2,u1andr2 v2−m12u1. In order to complete this step we need to show

that the vectorv2−m12u1is timelike.

Note that

v2−m12u12v22− |v2,u1|2, 3.2

so we just need to show that this number is positive. Sinceu12 _{1, this is equivalent to}

showing that

v22_{· u1}2_>_|v2,_u1|2_, _3.3

or, after multiplying both sides byr2

1, that

v22_{· v1}2_>_|v2_,_v1|2_. _3.4

Denote the coordinates of the vectorw1 bya1, . . . , ap, b1, . . . , bq, and the coordinates

ofw2byc1, . . . , cp, d1, . . . , dq. We have

p

i1

|ai|2₋

q

j1

bj2 p

i1

|ci|2₋

q

j1

dj21, 3.5

p

i1

aici−

q

j1

bjdj 0. 3.6

The coordinates of the vectorv1are given by

eλ1_a

1, . . . , eλpap, eμ1b1, . . . , eμqbq

3.7

andv2by

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The RHS of3.4now becomes as follows, using3.5:

⎛

⎝p

i1

e2λi|ai|2−

q

j1

e2μj_bj2

⎞ ⎠_·

⎛ ⎝p

i1

e2λi|ci|2−

q

j1

e2μj_dj2

⎞ ⎠

⎛ ⎝p

i1

e2λi−_e2μ1

|ai|2

q

j2

e2μ1−_e2μj_b

j2e2μ1

⎞ ⎠

·

⎛ ⎝p

i1

e2λi−_e2μ1

|ci|2

q

j2

e2μ1−_e2μj_d

j2e2μ1

⎞ ⎠_,

3.9

which is strictly greater than

⎛ ⎝p

i1

e2λi−_e2μ1

|ai|2

q

j2

e2μ1−_e2μj_b

j2

⎞ ⎠

·

⎛ ⎝p

i1

e2λi−_e2μ1

|ci|2

q

j2

e2μ1−_e2μj_d

j2

⎞ ⎠_,

3.10

which in turn, by Cauchy-Schwarz, is greater or equal than

p i1

e2λi−_e2μ1

aici

q

j2

e2μ1−_e2μj

bjdj

2

, 3.11

which is exactly|v1,v2|2_{, if we take into account}_3.6.

Now, we will similarly show thatukis timelike for anyk≤p. We have

vkk−1

1

mkurkuk, 3.12

wheremkvk,u.

This amounts to showing that

vk2 _>k−1

1

|vk,u|2. 3.13

Consider the subspaceV ⊂ n spanned byu1, . . . ,uk−1, which is the same subspace

that is spanned byv1, . . . ,vk−1. It is isomorphic to the subspaceW spanned byw1, . . . ,wk−1,

and the explicit isomorphism is given by multiplying the first coordinate bye−λ1_{, . . ., the}_pth

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Denote byykk−1₁mkuIfvk,yk0, then by definition ofykandmkthis would

imply thatvkis orthogonal to all theuifor 1≤i≤k−1, and thus allmk0 and, similar to

the casek1, the vectorukvk/|vk|is timelike as desired.

Thus, we can assume that

αkvk,yk

k−1

1

|mk|2_>_0. _3.14

Define

u √yk

αk. 3.15

This is an essentially unique element ofV up to a circle factorwith the property that:

vk,u ·uk−1

1

mku. 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 casek2 now applies verbatim, with the role of

w1played byw and the role of v2played byvk.

Thus we have established that the firstpcolumns ofsare timelike vectors, and that

r1, . . . , rpare positive real numbers. Due to the Sylvester’s law of inertia for quadratic forms,

the only way to complete this to apseudo-orthonormal basis with respect to·,·is to add

qspacelike vectors—therefore if we continue the Gram-Schmidt orthogonalization process,

then the lastqcolumns ofswill be spacelike as required.

4. Iwasawa-Type Decomposition

Let us introduce more notation. Denote byQadmandadmthe sets of admissible elements in

Qand, respectively. Recall thatQadmexpadm. 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∈be admissible, letg0∈G0, and let us find an explicit decomposition

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withg₀ ∈G0,a∈A, andn∈N. Notice that the symmetrization map yields

g₀−1exp2λg0n†a2n. 4.3

The right hand side provides the Gauss decompositionalso known as the triangular or the

LDV-factorizationof 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 ∈ Qadm are positive, then the eigenvalues of any

leading principal minor would be positive as well, and therefore the Gauss decomposition exists.

The diagonal entries ofa2are the ratios of the leading principal minors:

a2

ii

Δi

Δi−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_∈is admissible, then the whole orbit expλG0admits a global decomposition,

that is, expλG0⊂G0AN.

Another interesting property ofAN_admis that it is a multiplicative set.

Proposition 4.3. If two elementsb1 andb2 fromAN are admissible,b1, b2 ∈ AN_adm, then their

productb1b2is admissible as well.

Proof. Applying the dressing action, we can actually assume thatb2a∈Aadm. This follows

from the fact that forg ∈G0andb1, b2∈AN:

gb1b2

gb2

b1

, b1b2gbg₁ b2bg₂. 4.5

Applying the symmetrization map tob1a, we obtain a new matrixfab†₁b1a∈Q, which we

need to prove admissible.

Since the spaceAadmis clearly connected andais admissible, consider a pathatsuch

thata0 Id,a1 aandat∈Aadmfor 0< t≤1. Also denoteft atb†1b1at.

Assume, on the contrary, that f1 f is not admissible and let E ⊂ 0,1 be the

nonemptysubset defined by the property thatftfort∈ Eis not admissible. Then consider

τ infE.

We recall 3 that the space adm forms a convex cone in q, therefore a simple

infinitesimal computation can show that Qadm is a connected open subset of Q and any

element from the boundary of Qadm, such as fτ, has the property that its eigenvalues

remain real and, moreover, the lowest eigenvalueλpcorresponding to the timelike cone and

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exists a whole plane of eigenvectors containing vectors from both n

and n−. Thus it must

contain an eigenvector, which we denote byz, from the null cone.

For this eigenvector we have the following equation:fτzβz. Or, equivalently,

aτb†₁b1aτzβz. 4.6

Note that both aτ and b†₁b1, being admissible, map the timelike cone n into itself by

Lemma 2.1. Similar proof shows that they map the null cone minus the origin n₀ \ {0}inside

the timelike cone. Therefore, the elementaτb†₁b1aτalso maps n

0 \ {0}inside of n and

thusz cannot be its eigenvector.

In geometric terms, we have established the following characterization of the space of

admissible elementsQadm.

Proposition 4.4. An elements ∈Qis admissible if and only if it has real positive eigenvalues and

maps the closure of the timelike cone n n ∪ n₀ into the timelike cone n (plus the origin).

Notice that sinceafrom the above proof is admissible diagonal, the pseudohermitian

analogue of the Rayleigh-Ritz ratio

RA x_x†Ax_†_x 4.7

for the matricesf ands b₁†b1 will have the following properties. Letx ∈ n

be timelike,

then, as we saw earlier,ax is timelike as well with a bigger norm byLemma 2.1.

Thus, forx∈ n

, we have

Rasax x†asax

x†_x >

ax†_sax†

ax†ax Rsax. 4.8

It follows sinceamaps n

to itself, that

min

x∈ n

Rsx≤min

x∈ n

Rsax<min

x∈ n

Rasax. _4.9

From5, Theorem 2.1, it follows thatλ1asa> λ1s. Similar inequalities can be established

for other eigenvalues.

The notion of admissibility can be extended to the whole groupGSLn, . We say

thatg ∈ G is admissible, if it decomposes asg hb, with h ∈ G0 and an admissibleb ∈

AN_adm. Such a decomposition, if exists, is clearly unique. Note, moreover, thatg†g b†b.

The singular admissible spectrum ofgare the square roots of the spectrum ofg†g.Note of

warning: one should not defineQadmby inducing this definition fromG, but rather as we did

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Suppose that g1, g2 are two admissible elements from G such that g1 h1b1 and

g2 h2b2. Then we haveg1g2 h1b1h2b2 h1h₂b1bh₁2b2, where the powers mean the dressing

actions. This is possible, becauseb1is admissible. Thus we have established that the product

g1g2is admissible if and only if the productbh2

1 b2is such.

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

The Poisson geometry related to the group SU1,1was considered in detail in6. Here we

just recall several facts to illustrate the results of this paper. First of all, the spaceadmis the

following convex cone of matrices

z xiy

−xiy −z

:x, y, z∈, z

2₋_x2₋_y2_>_{0, z >}_0. _5.1

Next, we defineQadmexpadm. Consider an element

t1 m

−m t2

∈Q, 5.2

wheret1, t2∈, andm∈ satisfy the determinant conditiont1t2|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:

t1t2>2, t1>1. 5.3

Next, an element

r n

0 r−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 decompositionG⊃G0ANexplicitly. Let us

consider a general element of SL2, :

a b

c d

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The condition that this general element admits such a decomposition is simply|a|>|c|: a b c d ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ a

|a|2_{− |c|}2

c

|a|2_{− |c|}2

c

|a|2_{− |}_c|2

a

|a|2_{− |}_c|2

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠· ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

|a|2_{− |}_c|2

b− c

|a|2_{− |}_c|2

|a|2_{− |c|}2

a

0 1

|a|2_{− |}_c|2

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ . 5.7

Thus, it is quite easy to see that for any element g ∈ G and expλ ∈ Aadm, the element

expλg ∈G0AN. Explicitly, if

g

u v

v u

, expλ

eλ ₀

0 e−λ

, λ >0, 5.8

then

expλg

eλ_u _eλ_v

e−λv e−λu

5.9

is an admissible element ofG.

The statemement that the admissibility of two elementsb1 andb2 ofAN ⊂ SL2,

implies the admissibility of their product is also a short computational aﬀair.

References

1 P. Foth and J.-H. Lu, “Poisson structures on complex flag manifolds associated with real forms,”

Transactions of the American Mathematical Society, vol. 358, no. 4, pp. 1705–1714, 2006.

2 N. J. Vilenkin and A. U. Klimyk, Representation of Lie Groups and Special Functions. Vol. 3, vol. 75, Kluwer Academic Publishers Group, Dordrecht, The Netherlands, 1992.

3 K.-H. Neeb, Holomorphy and Convexity in Lie Theory, vol. 28 of de Gruyter Expositions in Mathematics, Walter de Gruyter & Co., Berlin, Germany, 2000.

4 P. Foth, “Polygons in Minkowski space and the Gelfand-Tsetlin method for pseudo-unitary groups,”

Journal of Geometry and Physics, vol. 58, no. 7, pp. 825–832, 2008.

5 P. Foth, “Eigenvalues of sums of pseudo-Hermitian matrices,” Electronic Journal of Linear Algebra, vol. 20, pp. 115–125, 2010.

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A Note on Iwasawa Type Decomposition

Research Article

A Note on Iwasawa-Type Decomposition

Philip Foth

1. Introduction

2. Admissible Elements

3. Gram-Schmidt Orthogonalization and Admissible Elements

4. Iwasawa-Type Decomposition

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

References

Submit your manuscripts at

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Mathematics

Complex Analysis

Optimization

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Function Spaces

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