Orbits of Lagrangian Subalgebras of the Double sl(2, R)

Orbits of Lagrangian Subalgebras ofthe Double slð2;RÞ

NICOLA CICCOLI and LUCIO GUERRADipartimento di Matematica e Informatica, Universita¤ di Perugia, Via Vanvitelli 1,06123 Perugia, Italy. e-mail: [email protected] [email protected]

(Received: 13 March 2000)

Abstract. We describe the variety of Lagrangian subalgebras of the Drinfeld double for anarbitrary bialgebra structure on slð2;RÞ.We determine the irreducible components and the orbitstructure under the natural action of the group SLð2;RÞ.

Mathematics Subject Classi¢cations (2000). primary 17B62; secondary 53D17.

Key words. Lie bialgebras Drinfeld double, Lagrangian subalgebras.

1. Introduction

A Poisson^Lie group G has a tangent Lie bialgebra ðg; dÞ from which the Drinfelddouble DðgÞ is constructed, which is a Lie algebra on g� g� for which the naturalinner product is invariant. The variety La of Lagrangian subalgebras of the doubleadmits a natural action of G. The construction provides a classifying space forPoisson homogeneous spaces, according to the following theorem of Drinfeld (in[2]).

Given a homogeneous space M of the group G, there is a bijective correspondencebetween covariant Poisson structures on M and equivariant mappings M ! La

sending x 7!Lx so that Lx \ g is the Lie algebra of the stabilizer of x. A recent paper[3] contributes a universal Poisson structure on La such that every induced mappingM ! La is a Poisson mapping.

Note that the theorem contains, as special cases, both the classi¢cation of af¢nePoisson structures on a Lie group (for which we refer to [5]) and the classi¢cationof symplectic homogeneous varieties. Related results concerning the classi¢cationof covariant Poisson structures are found in [6,7].

In this paper, we determine the irreducible components and the orbit structure ofthe variety La for an arbitrary bialgebra structure on g ¼ slð2;RÞ. The componentof the point g is a real P3, and the pointed P3

f�g admits an equivariant ¢brationover P2, the projectivized g under the adjoint action of G ¼ SLð2;RÞ. A completedescription of orbits is in Section 4. The component of g� is a quadric surface ina second P3, irreducible but possibly singular, where one orbit is at in¢nity andthe af¢ne piece is a second orbit, except possibly for one ¢xed point. This is shown

Geometriae Dedicata 88: 135^146, 2001. 135# 2001 Kluwer Academic Publishers. Printed in the Netherlands.

in Section 5. Finally, in Section 6, we determine the coisotropic orbits, those whichcarry a non-symplectic structure. The orbit decomposition is related to a naturalstrati¢cation according to the possible dimensions of intersections of Lagrangiansubspaces, which is presented in Section 3. A description of the irreduciblecomponents of La for a compact group G can be found in [3].

2. Preliminary Material

2.1. THE VARIETY OF LAGRANGIAN SUBALGEBRAS

Let T be a real vector space of even dimension 2n, endowed with a symmetric bilinearform h ; i of type ðn; nÞ. Up to isomorphism, we may identify T with V � V�, where Vis a vector space of dimension n, with the bilinear form hxþ x; yþ Zi ¼ xðyÞ þ ZðxÞ.We want to describe the variety L which parametrizes the Lagrangian subspacesof T .

The local coordinates in L are de¢ned in the following way. Let L;L0 be a pair of¢xed Lagrangian subspaces which are complementary in T . An open neighborhoodof L consists of all the Lagrangian subspaces which are complementary to L0. Iff : L! L0 is a linear map then the graph Lf ¼ fxþ f ðxÞ : x 2 Lg is complementaryto L0, and every complement to L0 is obtained in this way. Moreover the graphLf is Lagrangian if and only if f is h ; i-antisymmetric. Therefore in the open setof the Lagrangian complements to L0 the local coordinates are the points in the spaceof antisymmetric maps AðL;L0Þ. This gives to L the structure of a smooth realalgebraic variety of dimension nðn 1Þ=2. Moreover in this way L is an algebraicsubvariety of the Grassmannian GrnðT Þ.

We remark that L is naturally a homogeneous space of the group Oðn; nÞ, relativeto the stabilizer of a given Lagrangian subspace. As this stabilizer is a subgroupof SLð2nÞ it follows that L has two connected components. We will look at thecomponents in more detail in the special case of dimension n ¼ 3.

Now assume that T also carries the structure of a Lie algebra, for which the scalarproduct is invariant, i.e. the identity h½x; y�; zi ¼ hx; ½y; z�i holds. We want to describethe subvariety La of L which parametrizes the Lagrangian subspaces which aresubalgebras.

The local equations for La are obtained as follows. Let L be a ¢xed Lagrangiansubalgebra, and let L0 be a complementary Lagrangian subspace. In a neighborhoodof L the Lagrangian subspaces are of the form Lf with f 2 AðL;L0Þ. The condition forLf to be a subalgebra is that the identity

h½xþ fx; yþ fy�; zþ fzi ¼ 0

holds for every x; y; z 2 L.Equivalent is that the equality holds for every sequence x; y; z taken from a given

basis of L. In the special case of dimension n ¼ 3, because the scalar product is

136 NICOLA CICCOLI AND LUCIO GUERRA

invariant, it is enough to check the equation above for a single sequence x; y; z whichis a basis of L.

2.2. THE DRINFELD DOUBLE

On a Lie algebra g the structure of a bialgebra is given by a linear map d : g! gV

g,called cocommutator, such that d� is a Lie bracket on g�, and d is a 1-cocycle of gwithvalues in g� g with respect to the adjoint action.

If ðg; dÞ is a Lie bialgebra there exists a unique Lie algebra structure on g� g� suchthat g and g� are subalgebras (on g� the bracket d�) and for which the inner producthxþ x; yþ Zi ¼ xðyÞ þ ZðxÞ is invariant. The Lie bracket is:

½xþ x; yþ Z� ¼ ½x; y� ad�x ðZÞ þ ad�Z ðxÞ ad�x ðyÞ þ ad�y ðxÞ þ ½x; Z�

½x; Z� ½x; y�

where ad� denotes the dual adjoint action. The space DðgÞ ¼ g� g� endowed with theLie bracket above is called the Drinfel’d double of the bialgebra ðg; dÞ.

On a Poisson^Lie group G, identifying the tangent algebra gwith the space of rightinvariant vector ¢elds, a global 2-form Z : G! g ^ g is obtained from the Poissonbivector ¢eld. Its differential d : g! g ^ g is a cocycle, so there is a tangent bialgebraðg; dÞ.

If dðxÞ ¼ ½x; u� ^ u0 þ u ^ ½x; u0� is the coboundary of o0 ¼ u ^ u0 thenZðgÞ ¼ g u ^ g u0 u ^ u0 ¼ go0 o0. Here we write g u ¼ AdgðuÞ for the adjointaction of the group, and similarly go for the adjoint action on bivectors.

When g ¼ slð2;RÞ then every cocycle is a coboundary (vanishing cohomology) andevery coboundary is a cocommutator (vanishing Yang-Baxter equation). Explicitformulas are given in [1]. A classi¢cation of these bialgebra structures is foundin [4].

The action of G on the double DðgÞ is given by

gðxþ xÞ ¼ g xþ g1xþ g1xb ZðgÞ

where g1x ¼ Ad�g1 ðxÞ denotes the coadjoint action of g, and where the symbol bdenotes contraction of tensors.

2.3. THE ADJOINT ACTION OF SLð2;RÞ

We recall here some basic facts concerning the adjoint action of G ¼ SLð2;RÞ.Denote by H;X ;Y the usual basis in g ¼ slð2;RÞ. Apart from the null orbit, thenon-zero orbits of G in g are the ¢bres of det : g f0g ! R. The orbits of G inPrðgÞ, the projective space of g, are the three subsets

Dþ ¼ fdet > 0g D ¼ fdet < 0g D0 ¼ fdet ¼ 0g

represented by the vectors X Y ; H; X .

ORBITS OF LAGRANGIAN SUBALGEBRAS 137

The stabilizer of a point ½x� 2 PrðgÞ is the subgroup S½x� ¼ fg 2 G :

g x ¼ l x for some l ¼ lxðgÞg. Then lx : S½x� ! R� is a group homomorphism.Two elements which are equivalent under G have equivalent homomorphisms, whichhave the same image. The three basic stabilizers are

S½H� ¼k 0

0 k1

� �; k 6¼ 0

� �¼ R�

S½X Y � ¼ SOð2Þ;

S½X � ¼u v

0 u1

� �u 6¼ 0

� �¼ R� �R;

so for the elements X Y and H one ¢nds lðgÞ ¼ 1 identically, while for X one hasthe homomorphism lðgÞ ¼ u2. Summing up, we have the following

Remark. A point in Dþ [ D determines the trivial homomorphism lðgÞ ¼ 1, apoint in D0 determines an homomorphism such that imðlÞ ¼ R>0.

The adjoint action Ad ^ Ad of G on g ^ g is isomorphic to the adjoint action Ad.More precisely the Lie bracket v ^ w 7!½v;w� induces an isomorphism g ^ g$ g

which determines an isomorphism of actions Ad ^ Ad $ Ad. The non-zero orbitsare the ¢bres of the map D : g ^ g f0g ! R de¢ned by

Dðv ^ wÞ ¼ det ½v;w�

which is expressed asða2 þ 4bcÞ in terms of the coordinates ða; b; cÞ of v ^ w relativeto the basis X ^ Y ; Y ^H; H ^ X . The orbits are represented by the bivectorsa X ^ Y ; b H ^ ðX Y Þ; H ^ X .

The coadjoint action CoAd of G on g� is isomorphic to the action on bivectors. Thedeterminant map induces an isomorphism g ^ g$ g� which gives an isomorphismAd ^ Ad $ CoAd. Here one uses the fact that Ad passes through the special groupSLðgÞ. We will not make use of this.

3. Lagrangian Subspaces of Dimension 3

According to the possible values for the pair of dimensions

d ¼ dim L \ g d 0 ¼ dim L \ g�

the variety L is partitioned into strata

Lðd; d 0Þ

We also de¢ne larger strata

Lðd;Þ Lð; d 0Þ

where the blank means union over all dimensions. In the special case dim g ¼ 3 wewant to relate the strati¢cation to the components of L.


LEMMA 3.1. The strata Lð0; 0Þ; Lð2; 0Þ; Lð0; 2Þ are empty.Proof. In the open set d 0 ¼ 0 a Lagrangian subspace is of the form Lf corres-

ponding to some antisymmetric f : g! g�. Since Lf \ g ¼ ker f and since the rankof an antisymmetric f is even it follows that the dimension d is odd. Similarlyd ¼ 0 implies d 0 odd. &

LEMMA 3.2. The stratum Lð1; 1Þ is also empty.Proof. If L � gþ g� is a Lagrangian subspace, call M;M0 the images of the

projections of L into g; g�. If both dimensions d; d 0 are X 1 then there is a basisfor L of the form x; x0; yþ y0, where x 2 L \ g; x0 2 L \ g�. Then M is generatedby x; y and M0 is generated by x0; y0. It is easy to see that L is Lagrangian ifand only if the linear forms x0; y0 are zero on the vectors x; y, that is to sayM0 � AnnðMÞ. Therefore, if d 0 ¼ 1, then M has dimension 2 and thereforeM0 ¼ AnnðMÞ has dimension 1 which implies d ¼ 2. &

LEMMA 3.3. If L is a Lagrangian subspace of type ð1; 2Þ then L ¼M þM0 is a directsum of subspaces M � g and M0 � g� which are orthogonal in the duality. Moreover,M ¼ kerðf Þ and M0 ¼ imðf Þ for an antisymmetric map f : g! g� which is determinedup to proportionality. Similarly a Lagrangian subspace of type ð2; 1Þ is a direct sumL ¼M þM0 of orthogonal subspaces where M ¼ imðf Þ and M0 ¼ kerðf Þ for anantisymmetric map f : g� ! g.

Proof. Continuing the argument in the preceding proof, in the case d 0 ¼ 1. As y0 isa multiple of x0 then a basis for L is given by x; y; x0, hence L ¼M þM0. We havealready seen that M0 ¼ AnnðMÞ. De¢ne L ¼ g�=M0 so that L� ¼M. Anantisymmetric map f : g� ! g such that M ¼ imðf Þ and M0 ¼ kerðf Þ is inducedby an antisymmetric map L! L�. The space of these maps has dimension 1. &

We are now in a position to state the relation of the strati¢cation and thecomponents of the variety L, sketched in the following diagram:

ð1; 0Þ ð0; 1Þ. & . &

ð3; 0Þ ð1; 2Þ ð2; 1Þ ð0; 3Þg g�

G G0

where an arrow from a stratum to another says that the second is in the boundary ofthe former. The two pieces G;G0 are the components of L.

We have the identi¢cation Lð; 0Þ $ Aðg; g�Þ on the af¢ne piece, given by localcoordinates, and the bijection Lð1; 2Þ $ PrðAðg; g�ÞÞ on the boundary, given byLemma 3.3. Thus G is bijective to the projective space PrðV Þ whereV ¼ Aðg; g�Þ �R. Note that this projective space has a natural partition into strata,


a distinguished point in the af¢ne part and the space at in¢nity, and that the bijectionabove preserves the strati¢cations.

Similarly one sees that the variety G0, union of the af¢ne open set Lð0;Þ and theclosed stratum Lð2; 1Þ, is bijective to the projective space PrðV 0Þ whereV 0 ¼ Aðg�; gÞ �R, in a bijection which preserves the strati¢cations.

LEMMA 3.4. The natural bijection

PrðV Þ [ PrðV 0Þ ! G [ G0 ¼ L

is an isomorphism which preserves the strati¢cations.Proof. A direct veri¢cation by means of local coordinates is a long computation.

We just sketch a second proof. It is enough to prove that the map is a morphismin the direction PrðV Þ [ PrðV 0Þ ! L. The theorem then follows from a general fact,that a birational bicontinuous map onto a normal variety is an isomorphism. Itis enough to consider the restriction PrðV Þ ! L. Using the embeddingL,!Gr3ðgþ g�Þ one is reduced to prove that PrðV Þ ! Gr3ðgþ g�Þ is a morphism.Locally over small open subsets of PrðAðg; g�ÞÞ there are morphisms ½f �7!x½f �such that x½f � generates kerðf Þ. Over smaller open subset we may also assumethat two given vectors y; z are independent of every x½f �. Now consider thelocal morphism which sends ðf ; kÞ 2 V into the class of the 3-vectorx½f � ^ ðkyþ f ðyÞÞ ^ ðkzþ f ðzÞÞ. It is easy to check that for k ¼ 0 this gives the pointassociated to the Lagrangian kerðf Þ þ imðf Þ, and for k ¼ 1 this gives the pointassociated to Lf . &

4. In the Double slð2;RRÞ, Orbits in the Component C

We are going to describe the orbit structure of the variety of Lagrangian subalgebrasin the double DðgÞ for an arbitrary bialgebra structure on g ¼ slð2;RÞ. The startingpoint is the following:

LEMMA 4.1. The map L 7!L \ g is equivariant.Proof. The statement means that gL \ g ¼ gðL \ gÞ. An element of gL is

gxþ g1x0 þ g1x0bZðgÞ where xþ x0 2 L. It belongs to g if and only ifg1x0 ¼ 0 and hence if and only if x0 ¼ 0. So the element of gL is indeed gx wherex 2 L \ g. &

We ¢rst consider the piece contained in the component G. The subalgebra g is a¢xed point. So consider the stratum Lð1;Þ. Because of the lemma, the mapL 7!L \ g de¢nes an equivariant ¢bration p : Lð1;Þ!PrðgÞ which reduces theorbit structure of the total space to the orbit structure of ¢bres under the actionsof stabilizers. This is indeed the route that we are going to follow.


The ¢bre Lðh;Þ is identi¢ed to the space of antisymmetric maps f : h? ! g=h. Tothe map f is associated the Lagrangian L ¼ f xþ x0 : x0 2 h?; x ¼ f ðx0Þmod hg. If abasis x; y; z is given such that x is a generator of h, and if x0; y0; z0 is the dual basis,then there is a scalar k such that f ðy0Þ ¼ kz; f ðz0Þ ¼ ky modulo h. The correspond-ing Lagrangian L is generated by the vectors x; kzþ y0; kyþ z0.

THEOREM 4.2. All these Lagrangians are subalgebras. So the variety G is acomponent of La.

Proof. Let a Lagrangian subspace be given by means of a basis as before. Thecondition for a subalgebra is h½x; kzþ y0�;kyþ z0i ¼ 0, as follows from the lastparagraph in Section 2.1. After some computations one ¢nds that

h½x; kzþ y0�;kyþ z0i ¼ k h½x; y�; y0i þ k h½x; z�; z0i þ hx; ½y0; z0�i;

where

h½x; y�; y0i þ h½x; z�; z0i ¼ tr adx ¼ 0

because tr adx ¼ 0 for an element in a semisimple algebra, and moreover

hx; ½y0; z0�i ¼ 0

because h? is a subalgebra of g�, like all subspaces of dimension 2. In fact if thecoproduct d is the coboundary of the bivector o0 ¼ u ^ u0 then

hx; ½y0; z0�i ¼ hdðxÞ; y0 ^ z0i ¼ h½x; u� ^ u0 þ u ^ ½x; u0�; y0 ^ z0i

¼ hx ^ ½u; u0�; y0 ^ z0i ¼ hx; y0i h½u; u0�; z0i hx; z0i h½u; u0�; y0i ¼ 0

where the equality ½x; u� ^ u0 þ u ^ ½x; u0� ¼ x ^ ½u; u0� follows under the isomorphisma ^ b 7!½a; b� from the Jacobi identity. &

Every antisymmetric map f : h? ! g=h is obtained as contraction f ðx0Þ ¼ x0b �oowith a bivector �oo ¼ �uu ^ �vv in g=h ^ g=h. If L is the Lagrangian over h correspondingto �oo then the Lagrangian gL over h0 ¼ gh in the same way corresponds to a bivector�oo0 in g=h0 ^ g=h0. The action of g on Lagrangians is expressed as a mapg=h ^ g=h!g=h0 ^ g=h0.

LEMMA 4.3. The action of g is the quotient action of o 7!gðoþ o0Þ o0 where o0

is the bivector that gives the coproduct.Proof. Let L ¼ f xþ x0 : x0 2 h?; �xx ¼ x0b �oog. An element of gL is of the form

gxþ g1x0 þ g1x0b ZðgÞ ¼ yþ y0 where y ¼ gxþ g1x0b ZðgÞ and y0 ¼ g1x0.One needs to show that �yy ¼ y0b �oo0 where o0 ¼ gðoþ o0Þ o0. In fact moduloh0 one has gx ¼ gðx0b oÞ ¼ ðg1x0b goÞ ¼ y0b, go. Remind moreover thatZðgÞ ¼ go0 o0. It follows that y ¼ y0b ðgoþ ZðgÞÞ ¼ y0b o0 modulo h0. &

LEMMA 4.4. Action of the stabilizer SðhÞ on the ¢bre Lðh;Þ.


(1) If h 2 Dþ [ D then every g 2 SðhÞ stabilizes every L 2 Lðh;Þ.(2) If h 2 D0 then in the ¢bre there is a unique Lagrangian L0 which is a ¢xed point of

SðhÞ. The two components of the complement Lðh;Þ fL0g are two orbits.

Proof. Let g 2 SðhÞ. Then go ¼ ð1=lðgÞÞ �oo for any bivector �oo, where lðgÞ is theeigenvalue of h for g, introduced in Section 2.3. This is because the inducedendomorphism �gg of g=h ^ g=h satis¢es �gg �oo ¼ det �gg �oo and moreover1 ¼ det g ¼ lðgÞ det �gg as the adjoint action factors through SLðgÞ. Therefore theaction of g on the ¢bre is induced by the action o 7! ð1=lðgÞÞ ðoþ o0Þ o0. Point 1.In this case one has lðgÞ ¼ 1 for every g (see the remark in Section 2.3). Point 2. Oneneeds to solve the equation �oo0 þ �oo0 ¼ ð1=lÞ ð �ooþ �oo0Þwith l ¼ lðgÞ. Given �oo; �oo0 thereis a solution l > 0 if and only if �oo; �oo0 belong to the same hal£ine with origin �oo0 inthe line g=h ^ g=h, and then there is g such that lðgÞ ¼ l (same remark quoted above).The equation is satis¢ed for every g if and only if �oo0 ¼ �oo ¼ �oo0. &

THEOREM 4.5. Over the two orbits Dþ;D the ¢bration p admits equivarianttrivializations. Thus over each orbit there is a family of orbits parametrized bythe real line R. Over the orbit D0 there are three orbits.

Proof. Point 1. Because of the preceding lemma, there is an induced map

G=SðhÞ � Lðh;Þ !p1ðG hÞ

It is easily seen to be an isomorphism of varieties. Point 2. There is a bijection

orbits of SðhÞ in Lðh;Þ ! orbits over G h

which to an orbit SðhÞ � L with L 2 Lðh;Þ associates the orbit G � L, and converselyto an orbit V over G h associates the intersection V \ Lðh;Þ. The result then followsagain from the preceding lemma. &

The orbit structure of La implies the classi¢cation of Poisson homogeneous spacesof the group, according to Drinfeld’s theorem in the introduction. For instance onthe upper half plane X an homogeneous Poisson structure is given by an equivariantmorphism X ! La that covers the natural isomorphism X ! Dþ. The theoremabove then implies that the homogeneous Poisson structures on X form a familyparametrized by the real line R. Explicit formulas are given in [4].

5. Orbits in the Component C0

Here we describe the orbit structure of the second piece La \ G0. The variety ofLagrangian subalgebras La inherits from L a decomposition into strata

Laðd; d 0Þ

Consider the piece of type ð2; 1Þ. The map Lð2; 1Þ !Prðg ^ gÞ is an equivariantisomorphism.


LEMMA 5.1. A Lagrangian subspace L ¼M þM0 of type ð2; 1Þ is a subalgebra ifand only if M is a subalgebra.

Proof. If L is a subalgebra and if x; y 2M then ½x; y� 2 L and by de¢nition½x; y� 2 g whence ½x; y� 2M. Conversely if M is a subalgebra it is enough to provethat ½x; x0� 2 L for elements x 2M and x0 2M0. Now ½x; x0� ¼ ad�x ðx

0Þ þ ad�x0 ðxÞand one has to prove that ad�x0 ðxÞ 2M and ad�x ðx

0Þ 2M0. For y 2M one hashy; ad�x ðx

0Þi ¼ h½x; y�; x0i ¼ 0 because ½x; y� 2M and x0 2M0. For y0 2M0 one hashad�x0 ðxÞ; y0i ¼ hx; ½x0; y0�i ¼ 0 because ½x0; y0� ¼ 0. &

It is therefore induced an equivariant isomorphism

Lað2; 1Þ!S

onto the subvariety which parametrizes the subalgebras of dimension 2 in g.

PROPOSITION 5.2. The stratum Lað2; 1Þ is an orbit.Proof. Observe that in the list of orbits of the adjoint action on bivectors the only

one that contains subalgebras is the orbit of H ^ X . Therefore S is an orbit. &

Finally consider the action on Lð0;Þ, which is isomorphic to the space ofantisymmetric maps Aðg�; gÞ. Every antisymmetric map f : g� ! g is obtained ascontraction f ðx0Þ ¼ x0bo with a bivector o 2 g ^ g. Therefore the action of Gon Lagrangians is isomorphic to an action on g ^ g.

LEMMA 5.3. If the coproduct d is the coboundary of the bivector o0 then the actionof the element g is given by

o 7! gðoþ o0Þ o0

It follows that the orbits of the action are translates of the orbits of the adjoint action.Proof. To the bivector o corresponds the Lagrangian L ¼ fxþ x0 : x ¼ x0bog.

The Lagrangian gL is the set of vectors of type gðxþ x0Þ ¼ gxþ g1x0 þ g1x0bZðgÞ ¼ yþ y0 where y ¼ gxþ g1x0b ZðgÞ and where y0 ¼ g1x0. One needs to showthat if x ¼ x0b o then y ¼ y0bo0 where o0 ¼ gðoþ o0Þ o0. Sincegx ¼ gðx0b oÞ ¼ g1x0b go it follows that gðxþ x0Þ ¼ y0b ðgoþ ZðgÞÞ ¼ y0bo0. &

THEOREM 5.4. In the coordinates o the variety Lað0;Þ is de¢ned by the equation

Dðoþ o0Þ ¼ Dðo0Þ

where Dðv ^ wÞ ¼ det ½v;w�. If Dðo0Þ 6¼ 0 it is an orbit. If Dðo0Þ ¼ 0 the varietycontains a ¢xed point, corresponding to the bivector o0, and the complement isa second orbit.


Proof. Using the one-to-one correspondence between two-dimensional subspacesand bivectors up to scalars, in which the subspace generated by v;w correspondsto the bivector v ^ w. We may write o ¼ x ^ y and o0 ¼ y ^ z, where y belongsto the intersection of the two planes that correspond to the bivectors. Assume thatx; y; z are independent and call x0; y0; z0 the dual basis. The Lagrangian is generatedby the vectors z0; yþ x0;xþ y0. The condition for a subalgebra is written in theform

hx; ½x0; z0�i þ hy; ½y0; z0�i þ h½x; y�; z0i ¼ 0

which is symmetric to the equation found in the proof of Theorem 4.2 and is obtainedin the same way, applying Section 2.1. Computing coproducts we obtainh½x; y�; x0i þ h½y; z�; z0i þ h½x; y�; z0i ¼ 0 whence h½x z; y�; z0i þ h½x; y�; x0i ¼ 0 and

h½x z; y�; z0 þ x0i ¼ h½y; z�; x0i

De¢ne u ¼ x z and consider the basis u; y; z and the dual basis u0; y0; z0. Theequation is written as

h½u; y�; z0i ¼ h½y; z�; u0i

Since u ^ y ^ ½u; y� ¼ h½u; y�; z0i u ^ y ^ z we deduce the equivalent equation

u ^ y ^ ½u; y� ¼ y ^ z ^ ½y; z�

Using the formula

u ^ y ^ ½u; y� ¼ det ½u; y� H ^ X ^ Y

we deduce the ¢nal equation

det ½u; y� ¼ det ½y; z�

Then consider the case x; y; z dependent. If x ^ y ¼ 0 we have the subalgebra g� andthe equation above is satis¢ed. If x ^ y 6¼ 0 we may write y ^ z ¼ a x ^ y and thenwe may assume z ¼ a x. Introduce a basis x; y;w and the dual basis x0; y0;w0.Computing as before the condition for a subalgebra is written asð2aþ 1Þ h½x; y�;w0i ¼ 0 while the equation above becomes ð2aþ 1Þ det½x; y� ¼ 0.We know from Section 2.3 that they are equivalent. &

COROLLARY 5.5. The component of La contained in G0 is a projective quadric.Proof. Use the description of G0 as a projective space given in Section 3. The

stratum Lað0;Þ is the af¢ne quadric surface de¢ned by the equation in the theoremabove. The conic at in¢nity is therefore det½u; v� ¼ 0 and this is the equation ofLað2; 1Þ, according to Section 2.3. &

Remark that the quadric is irreducible. It is singular if and only if Dðo0Þ ¼ 0 and inthis case it has the unique singular point o0. Note that the condition for


nonsingularity of the quadric coincides with the condition for the correspondingbialgebra structure to be quasi^triangular (see [1]).

6. Coisotropic Orbits

Let L 2 Lð1;Þ. Recall that such a Lagrangian subspace is in fact a subalgebra. Weconsider the question whether there is some g 2 G such that gL 2 Lð1; 2Þ. If a Poissonhomogeneous space X is given by some map onto the orbit of L then the conditionabove is satis¢ed if and only if the Poisson structure on X is not symplectic (cf.[5]). We call these orbits coisotropic because X is then the quotient of the Poisson^Liegroup by a coisotropic subgroup.

The variety of coisotropic subalgebras is G � Lð1; 2Þ. It is invariant under G and thenatural map to PrðgÞ restricts to an equivariant ¢bration over the image. The dis-tribution of coisotropic orbits follows from the structure of these ¢bres.

There is a natural section PrðgÞ ! Lð1;Þ de¢ned from the bivector o0 whichgives the coproduct. For every h choose in Lðh;Þ the subalgebra correspondingto the bivector �oo0 2 g=h ^ g=h. The section is equivariant and the image L0 there-fore consists of three orbits, one over each orbit in PrðgÞ.

LEMMA 6.1. Let L 2 Lðh;Þ correspond to �oo 2 g=h ^ g=h. The orbit of L iscoisotropic if and only if �ooþ �oo0 2 jðGo0Þ where j : g ^ g!g=h ^ g=h is the naturalprojection.

Proof. The image gL in the ¢bre over g h corresponds to �oo0 whereo0 ¼ gðoþ o0Þ o0. It is of type ð1; 2Þ if and only if �oo0 ¼ 0 modulo g h and thisis if and only if oþ o0 g1o0 ¼ 0 modulo h. &

PROPOSITION 6.2. If o0 ¼ 0 the stratum Lð1; 2Þ is invariant, and is the union ofthree coisotropic orbits. Assume then o0 6¼ 0.

(1) Over Dþ the coisotropic orbits are as follows. If Dðo0Þ < 0 every orbit; if Dðo0Þ ¼ 0every orbit except for the orbit in L0; if Dðo0Þ > 0 the variety of coisotropic orbitsis ¢bred over Dþ and the ¢bre is the complement of an open interval in the real line.

(2) Over D every orbit is coisotropic.(3) Over D0 the coisotropic orbits are as follows. If Dðo0ÞW 0 every orbit; if Dðo0Þ > 0

every orbit except for the orbit in L0.Proof. For o0 ¼ 0 the proof is immediate. Consider the case o0 6¼ 0. We want to

determine the coisotropic subalgebras over a ¢xed h, using Lmma 6.1, and we onlyneed to consider one point h for each orbit in PrðgÞ. The orbit Go0 is the quadricsurface which in coordinates is given by the equation ða2 þ 4bcÞ ¼ u whereu ¼ Dðo0Þ, minus the origin if u ¼ 0 (Section 2.3). The ¢bres of the linear projectionj are the translates of the plane h ^ g.

Choose h ¼ hX Y i as a point in the orbit Dþ. The translates of h ^ g are theplanes b c ¼ v. Then v is a coordinate in the quotient line


R ¼ g=h ^ g=h ¼ g ^ g = h ^ g. The quadric a2 þ 4bc ¼ u intersects the planeb c ¼ v at some point different from the origin if and only if v belongs to the imagejðGo0Þ. So by elementary calculations we ¢nd that jðGo0Þ is:R if u < 0,R f0g ifu ¼ 0, the subset v2 uX 0 if u > 0. Point 1 in the statement follows easily.

Choose h ¼ hHi in the orbit D. The ¢bres of j are the planes a ¼ v and withsimilar calculations one ¢nds that jðGo0Þ ¼ R for every u. Finally considerh ¼ hXi in the orbit D0. The ¢bres of j are the planes b ¼ v and jðGo0Þ isfound to be: R if uW 0, and R f0g if u > 0. This proves points 2,3 in thestatement. &

References

1. Aminou, R. and Kosmann-Schwarzbach, Y.: Bige¤ bres de Lie, doubles et carre¤ s, Ann. Inst.H. Poincare¤ Phys. Te¤ or. 49 (1988), 461^478.

2. Drinfeld V. G.: On Poisson homogeneous spaces of Poisson^Lie groups, Theoret. Math.Phys. 95 (1993), 524^525.

3. Evens, S. and Lu, J.-H.: On the variety of Lagrangian subalgebras, Preprintmath.DG/9909005.

4. Leitenberger, F.: Quantum Lobachevskij planes, J. Math. Phys. 7 (1996), 3131^3140.5. Lu, J.-H.: Multiplicative and af¢ne Poisson structures on Lie groups, PhD thesis,

University of California, Berkeley, 1990.6. Lu, J.-H. and Weinstein, A.: Poisson-Lie groups, dressing transformations and Bruhat

decomposition, J. Differential Geom. 31 (1990), 501^526.7. Sheu, A. J. L.: Quantization of the Poisson SUð2Þ and of its Poisson homogeneous space ^

the 2-sphere, Comm. Math. Phys. 135 (1995), 217^232.


Orbits of Lagrangian Subalgebras of the Double sl(2, R)

Documents

Transcript of Orbits of Lagrangian Subalgebras of the Double sl(2, R)