Geometry of Certain Lie-Frobenius Groups

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Preprint submitted on 18 Jun 2005

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Jean Michel Dardie, Alberto Medina, Hassène Siby

To cite this version:

Jean Michel Dardie, Alberto Medina, Hassène Siby. Geometry of Certain Lie-Frobenius Groups. 2005.

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ccsd-00005449, version 1 - 18 Jun 2005

Jean Mihel Dardié, Alberto Medina and Hassène Siby

Abstrat. Let be G_n,p(K) = M_n,p(K) ⋊GL(Kⁿ)^, K = R or C, the semi-diret produt of the additive group of matries Mn,p(K) ^by ^the

group GL(Kⁿ) ^where ^the ^ation ^is ^done ^by multipliation of matries.

If n = kp^,p ≥ 1 ^the ^Lie ^group Gn,p(K) âdmits ân êxat ^left învâriant

sympletiform[10℄.Westudythegeometryofthissympletimanifold.

If n > p ⁽ ^resp. n = p ⁾ ^we ^prove ^that G_n−p,p(K) ^(resp G_n−1,1(K)

) is the sympleti redution of the sympleti orthogonal (Mn,p(K))^⊥

of M_n,p(K) ⁱⁿ G_n,p(K) ând ^reiproally ^that G_n,p(K) îs â ^sympleti

double extension,inthesenseof[6℄, ofG_n−p,p(K) ⁽^resp. ^ofG_n−1,1(K)^).

Moreover we show that G_n,p(K) ^admits ^two ^left ^transverse ^Lagrangian

foliations (with ane and losed leaves) . Consequently there exists (a

leftinvariant)aanonialtorsionfreesympletionnetiononGn,p(K)^.

Key words :

Exat sympletiLiegroup,AneLiegroup, SympletiRedution,Sym-

pleti double extension.

Introdution

The groupG_n,1(K) =Af f(Kⁿ) ^where K =R or Cadmits left invariant sympleti strutureswhihareall exat beause H²(af f(Kⁿ),K) = 0^([9℄).

To give an invariant sympleti form on G_n,1(K) ^is ^to ^give ^a ^linear ^form α

on the Lie algebra af f(Kⁿ) ^suh ^that^the ^2-obound δα ^is ^non degenerate.

That isto sayα îsâ ^point^with ânôpenôrbit ûnder^theôadjoint âtionôf G_n,1(K)^.^These ^points^have ^been haraterized in[9℄ within the framework of study of ane group representations and in [3℄ where is arried on the

study of left invariant sympleti struturesof ane groupinitiated in[4℄.

In [10℄, it showed that oadjoint ation of Lie group semi-diret produt

of M_n,p(K) ^and GL(Kⁿ) ^by ^matriial ^produt, ^where M_n,p(K) ^indiate ^the

additive groupof(n×p)^-matries, âdmitsôpen ôrbitsîfônlyîfn=kp^.^Wê

extend to these groups denoted Gn,p(K) ^the ^result ^obtained ⁱⁿ ^[3℄ ^for ^the

lassial anegroup:existeneofaunique sympletistruture upisomor-

phism (Theorem2.7),existeneoftransversesympletifoliations(Theorem

2.6),aswellasofaLagrangian bi-foliationwithlosedleaves(Theorem3.2).

Suh a pair of Lagrangian foliations is important in the quantization pro-

edure (polarization) and implies the existene of a anonial(torsion free)

sympleti onnetionon Gn,p(K)^.^The ^natural ^leftâtion ôf Gn,p(K) ônît-

self beinghamiltonianwe anprovide(Theorem 2.1)asimilarofstruture's

theorem ofsimplyonneted sympletiLie group from[5℄.

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Inthe third partwestudy upthebrationsofthis theoremforexpliitsug-

gestaonstrutionof Gn,p(K), dα⁺₁

whenthegeneralizeddoubleextension

of [5℄ don'tapply inthis ase.

ToompletethisintrodutionreallthatasympletiLiegroup(G, ω⁺)^has

an ane struture given bythe left invariant onnetion ( torsion free and

zero urvature )∇^dened ^for^all x, y, z∈Tε(G) ^by ^: ω⁺(∇⁺_x+y⁺, z⁺) =−ω⁺(y⁺,[x⁺, z⁺])

where x⁺ ^denotes ^the ^leftînvariant ^vetorêldâssoiated ^to x^.

Then one says thatthe pair (G,∇⁺) îsân âne ^Lie^group. În^this âse ^the

produtonG^,xy = (∇⁺_x+y⁺)îs^withâssoiator^leftînvariantând^veries^the

ondition xy−yx= [x, y]^.^This^struture ^plays^a ^entral ^ruleⁱⁿ^this ^study.

1. Left invariant sympleti strutures on the Lie groups Gn,p

In what follows G_n(K) ^denotes ^the^lassial ^ane^group ^where K=Ror

C and Gn(K) îs îts ^Lie âlgebra. ^By ânalogy^, ^we ^denote Gn,p(K) ^the ^group

semi-diret produtM_n,p(K)⋊Gl(Kⁿ) ^withn=kp, p≥1 ^andG_n,p(Kⁿ)^,^its

Liealgebra. ObviouslyGn,1(K)^is^isomorphi^to Af f(Kⁿ)^.Considering that

H²(af f(Kⁿ),K) = 0 ^(see ^[3℄) âny ^left învâriant âlternate ^2-form îs învari-

antly exat. Thisresultbeomesgeneral to Gn,p(K) ⁱⁿ^the^following ^way^:

1.1 Lemma.Every left invariant sympletiformon G_n,p(K) ^is^exat.

Proof.Ifk=p= 1^;G1,1 îsîsomorphi ^toaf f(K)ând ^the^result^follows.

Assume that k ^(or p⁾ ^is ^greater ^than ^1.Let ^be ω ∈ Z²(G_n,p;K)^, G_n,p = Mp,n(K)⋊gl(n)^.^This^means ^that ^we^have

I

ω([a, b], c) = 0 (^*) ^for ^every a^,b^,c ⁱⁿ G_n,p.

If we take a= (x, u)^,b= (y, v) ^andc= (0, Id)

(*) implies ω(x, y) = 0 ^for ^every x, y ∈ Mn,p^, ^and onsequently ω(x, b) + ω(a, y) =ω(I, uy−vx+ [u, v])^.

But this isequivalent to

ω(a, b) =ω(I, ux−vy)

=−δβ(a, b) ^with β =ω(I,·)

N.B In fatfor all Lie algebra G ^having ^an ^element a∈ G ^suh ^that ada

is projetor,wehave H²(G,K) = 0^.

Remark 1.The mapping

Mp,n(K)×gl(n)−→(Mn,p(K)×gl(n))^∗ (H, M)7−→α_(H,M₎

given by

α_(H,M₎(N, V) =tr(N.H) +tr(M.V)

is alinear isomorphism.

Inthefollowing,thedualspaeG_n,p^∗ ôfGn,pîsîdentied^withMp,n(K)×gl(n)^.

Togiveaninvariantlosed 2^-formônGn,p(K)îtîsêquivalent^to^giveâ^linear

form α ôn G_n,p(K) ^. ^The 2^-form dα⁺^, ^where α⁺ ^denotes ^the ^left învâriant 1^-form ^dened ^by α îs ^sympleti îfând ônly îf α ^has ân ôpen ôrbit ûnder

the oadjoint representation.

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Using the Remark1 and thenatural embedding of G_n,p ⁱⁿGL(K^n+p) ^we

an show

1.2 Lemma.Theoadjointrepresentations ofGn,p^andGn,p^are^given^by

the following formulas:

(i) ad^∗_(0,u)(H, N) = (−H,[u, N])

(ii) ad^∗_(x,0)(H, N) = (0, x.H)

(iii) Ad^∗_(0,U₎(H, N) = (H.U⁻¹, U N U⁻¹)

(iv) Ad^∗_(X,Id

Kn)(H, N) = (H, N +X.H)

This implies

Ad^∗_(X,U)(H, N) = (H.U⁻¹, U N U⁻¹+XHU⁻¹)

Consequently the oadjoint orbit Ad^∗_G

n,p(α) ^of α = (H0, N0) ^with H0 = (0, I_p)∈M_p,n ^and

N₀ =





0 0 0

Ip 0 0 0 I_p 0





isopen.DenotebyH₀^the(p, n^)-matrix^whosep×p^bloksâreâll^nullêxept

forthelast,whihistheidentityofK^p anddenotedbyN0 ^the(n×n)^-matrix

whose p×p^bloksâreâll^nullêxept^thesub-diagonalswhihareId_K^p^.^The

previous formulas allow us toverifythattheorbit of (H₀, N₀) ^is ^open ^sine

the isotropysubalgebrais trivial.

2. SympletiRedution - Left invariant Sympleti foliation on

Gn,p(K)^.

Denote byω⁺=dα⁺^,α= (H, N)∈ G_n,p^∗ â^leftînvariant ^sympleti^formôn Gn,p^.

The ation ofG_n,p(K)^on G_n,p(K) ^given ^by

LG:Gn,p(K)×Gn,p(K)−→Gn,p(K), (τ, σ) 7−→τ σ( ^produtⁱⁿGn,p)

is a sympleti ation. Morever L_G ^is ^a Hamiltonian ation; a momentum mapping for the ation isgiven by

µ:G_n,p(K)−→ G_n,p(K)^∗ σ7−→µ(σ) :x7−→ hα⁺(σ), x⁻(σ)i

where x∈ Gn,pând x⁻ îs ^the^right învariant ^vetorêldâssoiated ^to x^.

ThesubgroupH:=Mn,pîs^(totally)îsotropi^forω⁺^beauseHîsânâbelian

Lie group. Morever L_H : H ×G_n,p −→ G_n,p îs â hamiltonian ation and a momentum mapping for L_H îs^given ^by^:

m:Gn,p−→L(H)^∗ σ7−→µ(σ)_|L(H)

where L_H îs^the^Lieâlgebra ôf H^.^Wê ârrive ât ^the^following ^theorem^:

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2.1 Theorem .Let (G_n,p, d((H, N)⁺) ^dened ^as^above.^Then

1.m⁻¹(H) îsâ ^losed ^subgroupôf G_n,p ândH ⊂m⁻¹(H)

2.The anonialexat sequene ofLie groups

(2) {ǫ} → H →m⁻¹(H)→m⁻¹(H)/H → {ǫ}

is split. Itis alsoan exat sequeneof aneLiegroups.

3.The reduedsympleti Liegroup m⁻¹(H)/H ^is^isomorphi^to ^: G_n−p,p(K) si n > p

G_p−1,1(K) si n=p >1

4.In the prinipal bundle

(3) m⁻¹(H)−→ⁱ Gn,p(K)−→^m Θ

where Θ îs ^the ^set ôf ^matries ôf ^rank p ⁱⁿ M_n,p^∗ ≡ M_p,n^, ^the ^ber îs ân

ane Lie subgroup and m îs âne ^relative ^to ^the ûsual âne ^struture ôf Θ⊂K^np.

We needthe following lemma forwhih theproof isobvious.

2.2 Lemma .The mapping m^is ^a ^surjetivesubmersion.

Furthermore,

(4) m((X, T)) =H.T⁻¹ ^where (X, T) ^is ⁱⁿ G_n,p(K)

Proof. The formula (4) is a diretonsequene of the denition of m ^and

theLemma 1.2.Thus itislear thatmîsâ^surjetive ^submersionôn ^the^set Θ ôf^matriesôf ^rankp ôf Mn,p(K)^.

Proof of the theorem 2.1.

Formula (4) implies that m⁻¹(H) = {(X, T) ∈ Gn,p(K)/HT⁻¹ = H} ^is ^a

(losed) subgroup of Gn,p(K) ^whih ôntains H^. ^Morever ^the ^fator ^group m⁻¹(H)/H ân ^be îdentied ^with {(0, T) ∈Gn,p/HT⁻¹ =H}^.^Thus ⁽²⁾ îs

an splitsequeneof Liegroups. Ontheotherhand,sine H^is ^ommutative

and ω⁺îsêxat,ît^follows^thatL(H)⊂L(H)^⊥ ândâstraightforward shows that L(m⁻¹(H)) =L(H)^⊥^.

Moreverfrom theformula

ω(xy, z) =−ω(y,[x, z])

deningtheleftsymmetriprodutinGn,p(K)^(see^formula⁽¹⁾⁾^it^turns^out

that L(H)^⊥ îs â ^left ^symmetri ^subalgebra ôf Gn,p(K) ând ^that L(H) îs â

two-side idealof L(H)^⊥^.Consequently (2) isanexat sequeneofane Lie groupsi.eH^,m⁻¹(H)^,m⁻¹(H)/Hâreâne^Lie^groupsând^theappliations of formula (2)areane.

On the other hand a list of the elements of the matrix group {(0, T) ∈ G_n,p(K)/HT⁻¹ = H} âllows ^to ôbserve ^that ^the ^group m⁻¹(H)/H îs îso-

morphi to the group Gn−p,p(K) îf n > p ând îsomorphi ^to Gn−1,1(K) îf n=p^.^This^proves ^statement ^3.

Now,weneed toshowthatthemanifoldΘîsêndowed ^withân âne^stru-

ture whihmakesthe momentum mapping ane.

Let F ^be ^the ^subbundle ôf T G_n,p(K) ^tangent ^to ^the L_H^-orbit ând F^⊥ îts

sympletiorthogonal.DenotebyF ^andF^⊥^the^assoiated^foliations^respe-

tively.SineHîs^normalⁱⁿG_n,p(K)^,^the^foliationF^⊥îs^denedêither^by^the

left invariant form on Gn,p(K) ^given ^by η_j^′ := i(e⁺_j )ω⁺ ^where (ei)^'s ^form ^a

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basisofL(H)ândi^denote^theînterior^produt,ôr^by^the^losed ^forms^(thus

exat)ηj :=i(e⁻_j )ω⁺^.^Obviously^theηj ^are^basi ^for^the^bration.^the^forms

¯

ηj ^whihâre^the^pro^jetionsôfηj ^bym ^deneâ^loalparallelismon Θ^.^This

parallelism isglobal and ommutative beausetheη¯j âreêxat.^Hene m îs

ane.

Letα₁ândα₂^be^the^linear^formsônG_n,p(K)^denedrespetivelybyα₁(x, u) = tr(H.x)ând α₂(x, u) =tr(N.u)^.^We^get ^:

2.3. Theorem. ker(dα₁) ^and ker(dα₂) ^are supplementary sympleti Lie subalgebras of(Gn,p(K), dα)^.^So^they^determine ^two^tranverse^sympleti^fo-

liations leftinvariant on G_n,p(K)

Proof. Obviously ker(dαi) îs â ^Lie ^subalgebra ôf Gn,p(K)^. În âddition ^the

subspaes ker(dαi)^,i= 1,2 ^areⁱⁿ^diret^sum ^beause ^ker(dα₁)T

ker(dα₂)=

{0}^.

If we take α = (H₀, N₀) ^we ^diretly ^observe ^that ^we ^have ^dim(ker(δα₁))=

p²(k−1)k ând^dim(ker(δα₂))= 2p²k^.^Weêxtends^these^resultsâbout^the^di-

mensionforallα∈ G_n,p^∗ (K)^withôpenôadjointôrbit.onsequentlyker(δα1)

and ker(δα₂)^are^sympleti^Lie subalgebras of(Gn,p(K), dα)^.

Let C(N₀) ^be ^the ^subalgebra ^ofgl(Kⁿ) ^given^by^:





 A₀ A₁ ^.^.^.

.

. .

.

. .

.

A_k−1 · · · A₁ A₀







2.4. Sholie. The Lie algebra ker(dα₁) îs îsomorphi ^to ^the ^Lie âlgebra G_n−p,p(K) îf n > p ând G_p−1,1(K) îf n= p ^while ker(dα₂) îs îsomorphi ^to

the semi-diret Mn,p(K)⋊C(N₀) îf n > p ând ^to ^the ^semi-diret ^produt M_p,p(K)⋊C(N₀) îfn=p^.

The following result is the innitesimal version of Theorem 2.1. It gives a

more preise andomplete statement of Theorem 2.1.

2.5 Proposition. With the notations of Therem 2.1., if I = L(H)^, ^the

anonial sequeneof vetoriel spaes

0→I →I^⊥→I^⊥/I →0

is a split exat sequene of Lie algebras. It is also an exat sequeneof left

symmetri algebras.

Furthermore the Lie algebra Gn,p(K) ^deomposes âs â ^diret ^sum ôf ^Lie

subalgebras I^⊥ ^andC(N₀) ^.

2.6 Theorem. The sympleti Lie group (Gn,p, dα⁺) ^is ^endowed ^with

two transversal left invariant sympleti foliations whose leaves are ane

submanifolds ofGn,p(K)^.

The following assertion speies the numberof open orbitsin G_n,p^∗ (K) ^as

wellasthe left invariant sympletistrutureson G_n,p(K)^.

2.7 Theorem.

a. There existtwo open orbitsof the oadjoint representation of Gn,p(K) ^if K=Rand only one ifK=C

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b. Up isomorphism there is only one left invariant sympleti struture on

G_n,p î.eîfω ândω^′ âre^two ^leftînvariant ^sympleti^formsônG_n,p(K)^,^then

there existsan automorphismϕ^of ^Lie^algebra Gn,p(K) ^suh ^that^: ωε(., .) =ω^′_ε(ϕ., ϕ.).

Thefollowing lemmassetupthemainstepsofthedemonstrationofThe-

orem 2.7. Thislemmaallow toount the Ad^∗_G

n,p

-orbits

2.8 Lemma. If Orb_(H,M₎ îs ^the ôadjoint ôrbit ôf (H, M) ∈ G_n,p^∗ ≡ M_p,n(K)×gl(n) ^then Orb_(H,M₎ ^has ân êlement (H₀^′, M₀^′) ∈ G_n,p^∗ ^suh ^that H₀^′ = (0,· · ·, A_p)∈M_p,n^.

Morever, if Orb_(H,M) îs ôpen, ^then H₀^′ ân ^be ^taken âsH₀^′ = (0,· · · , Ip) = H₀^.

Proof. It sues to remark that there exists U ∈ GLo(Kⁿ) ^suh ^that HU⁻¹ = H₀ ^, ^that îs ôbvious îf ^we ^look ât H âs^the ^matrix ôf â ^mapping

from Kⁿ into K^p and U⁻¹ âs^the ^matrix ôf â ^hange ôf ^basisⁱⁿ Kⁿ. Then the lastformulaoflemma 1.2showthe rstassertion.

Nowsupposethat Orb_(H,M₎=Orb_(H^′

0,M₀^′) îsâôpenôrbit. ^Thisîmplies^that

wehave

(x, u)∈ Gn,p;ad^∗_(x,u)(α) = 0⇒(x, u) = 0

In partiular

ad^∗_(x,0)(H₀^′, M^′) = 0⇒x= 0

In otherwords

(∗∗) tr(xH₀u) = 0, u∈gl(n)⇒x= 0

with H0 = (0, Ap)

But astraight alulationshows that(**)impliesAp ^is ⁱⁿ^vertible.

Finally there existsU ∈GL_o(Kⁿ) ^suh^that H₀^′U⁻¹ =H₀^,^but ^this ^relation

xonlythelast diagonalblokofU ^to ^the^valueA⁻¹^.^Therefore^ifk≥2^we

able to take an other diagonal blok egals to detA⁻¹.Id_K^p ^and ^ompleting

the diagonalebythe 1^,^to^onstrut ^a^suh^matrix U ⁱⁿ SL(Kⁿ)^.

2.9Lemma.AnopenorbitOrb_(H₀_,M)ôntainsônlyôneêlement(H₀, M^′)^,

where the blokdeomposition ofM^′ ^an ^be ^written ^: M^′ =

M₁ 0 H₁ 0

with (H1, M1)∈ G_n−p,p^∗ ^.

Proof.Beause

Ad^∗_(X,Id)(H₀, M) = (H₀, M +X.H)

it islear thatthere isonly one X∈M_p,n ^suh ^thatM^′ =M +M H₀^.

2.10 Lemma.Thelinearforms(H₀, M^′)ând(Ho, P^′)ônGn,p^withM^′ = (H1, M1) ând P^′ = (K1, P1)^, ^dened âs ⁱⁿ ^Lemma ^2.9, ^belong ^to ^the ^same Ad^∗_G

n,p

-orbitifandonly if(H₁, M₁) ^and (K₁, P₁) ^areⁱⁿ^the^sameAd^∗_G

n−p,p

-

orbit.

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Proof. (H₀, M ) ând (Ho, P ) âre ⁱⁿ ^the^same ôrbit îfând ônly îf ∃U ∈ GL_o(Kⁿ) ^suh^that U M^′U⁻¹ =P^′ ândH₀U⁻¹ =H_o^.

The seond onditionimplies

U =

U^′ 0 X1 Id

, with U₁ ∈GL₀(K^n−p)

OnotherhandU M^′U⁻¹=P^′ⁱⁿgl(Kⁿ)^is^equivalent^toAd^∗_(X₁_,U₁₎(H₁, M₁) = (K₁, P₁) ⁱⁿG_n−p,p(K)

Proof of the theorem 2.7.

Following the previous Lemmas we have

ardinal

nopen Ad^∗_G

n,p−orbito

= ardinal

nopen Ad^∗_G

p,p−orbito

On the other hand,using similar arguments asthose ofthe lemmas we an

showthat

ardinal

nopen Ad^∗_G

p,p−orbito

=ardinal

nopen Ad^∗_G_1,1−orbito

=

2 ^if K=R 1 ^if K=C

If K=C thereisonly one sympletistruture onGn,p^.

In the aseK=R,itfollowsfrom the lemmasthateveryAd^∗_G

n,p

-openorbit

ontains an element (H₀, N₀)^where

N₀ =





 0 A_p ^.^.^.

Ip

.

. .

.

I_p 0







withA_p ⁱⁿ^vertible

Neverthelesstwo matriesasN₀ âreônjugate ^byân êlement P ôf GL(n)^.

Howeverthe mapping

G_n,p^∗ −→ G_n,p^∗ , (g, M)7−→(P^t⁻¹g, P⁻¹M P)

is dualof themapping

θ:G_n,p−→ G_n,p (x, N)7−→(P⁻¹x, P⁻¹N P)

and these later isan automorphismof theLiealgebra G_n,p^.

Consequently ifω⁺₁ ^,ω₂⁺ âre^two ^leftînvariant ^sympleti^forms ônGn,p^,^we

have :

ω⁺₁ =θ^∗(ω₂⁺).

Remark2.NotiethattheonlyoneleftinvariantanestrutureonG_n,p

is givenby(H0, N0)^.

The following resultisan important onsequene oftheprevious study.

2.11Proposition.TheidentityomponentofGn,p^is^dieomorphi^to^an

open Ad^∗_G

n,p

-orbit. Consequently G_n,p ^and

α ∈ G_n,p^∗ ;α ^isPoisson-regular are dieomorphi.

Proof.(Byindution)Wemustprovethattheorbitalmappingin(H₀, N₀) (Gn,p)o−→ G_n,p^∗ , (X, U)7−→(H0U⁻¹, U N0U⁻¹+XH0U⁻¹)