NinthInternationalConferenceon Geometry,IntegrabilityandQuantization June8–13,2007,Varna,Bulgaria IvaïloM.MladenovandManueldeLeón,Editors SOFTEX,Sofia2008,pp1–21 8 0 0 2 n GEOMETRY AND TOPOLOGY OF COADJOINT ORBITS OF a J SEMISIMPLE LIE GROUPS 9 2 JULIA BERNATSKA†‡, PETRO HOLOD†‡ ] T R Department of Physical and Mathematical Sciences, National University of . † h ’Kyiv-Mohyla Academy’ 2 Skovorody Str., 04070 Kyiv, Ukraine at Bogolyubov Institute for Theoretical Physics of the National Academy of Sciences ‡ m of Ukraine 14-b Metrolohichna Str., 03680, Kyiv, Ukraine [ 2 Abstract. OrbitsofcoadjointrepresentationsofclassicalcompactLiegroups v havealotofapplications. Theyappearinrepresentationtheory,geometrical 3 quantization,theoryofmagnetism,quantumopticsetc.Asgeometricobjects 1 the orbits were the subject of much study. However, they remain hard for 9 calculationand application. We proposesimple solutionsfor the following 2 . problems: an explicit parameterization of the orbit by means of a general- 1 ized stereographic projection, obtaining a Kählerian structure on the orbit, 0 8 introducingbasistwo-formsforthecohomologygroupoftheorbit. 0 : v i 1. Introduction X r a Orbits of coadjoint representations of semisimple Liegroups are an extremely in- terestingsubject. Thesehomogeneousspacesareflagmanifolds. Remarkable,that thecoadjointorbitsofcompactgroupsareKählerianmanifolds. In1950sA.Borel, R.Bott,J.L.Koszul,F.Hirzebruchetal. investigated thecoadjoint orbitsascom- plexhomogeneousmanifolds. Itwasproventhateachcoadjointorbitofacompact connected Lie group G admits a canonical G-invariant complex structure and the only (within homotopies) G-invariant Kählerian metrics. Furthermore, the coad- jointorbitscanbeconsideredasfibrebundleswhosebasesandfibresarecoadjoint orbitsthemselves. Coadjointorbitsappearinmanyspheresoftheoreticalphysics,forinstanceinrep- resentation theory, geometrical quantization, theory of magnetism, quantum op- tics. They serve as definitional domains in problems connected with nonlinear integrable equations (so called equations of soliton type). Since these equations 1 2 JuliaBernatska†‡,PetroHolod†‡ have a wide application, the remarkable properties of coadjoint orbits interest not onlymathematicians butalsophysicists. Itshouldbepointedoutthatmuchofourmaterialis,ofcourse,notnew,butdrawn from various areas of the mathematical literature. The material was collected for solvingthephysicalproblembasedonaclassicalHeisenbergequationwithSU(n) asagaugegroup. Theequation describes abehavior ofmagnetics withspins>1. The paper includes an investigation of geometrical and topological properties of the coadjoint orbits. We hope it fulfills a certain need. We would like to men- tion that we have added a number of new results (such as an explicit expression for a stereographic projection in the case of group SU(3) and improving the way of its computation, the idea of obtaining the Kählerian potential on an orbit, an introduction ofbasistwo-formsforthecohomology ringofanorbit). Thepaper isorganized asfollows. Insection 2werecall thenotion ofacoadjoint orbit, propose aclassification oftheorbits, anddescribe theorbitasafibrebundle over an orbit with an orbit as a fibre. Section 3 is devoted to a generalized stere- ographic projection from a Lie algebra onto its coadjoint orbit, it gives a suitable complexparameterization oftheorbit. Asanexample,wecomputeanexplicitex- pression for the stereographic projection in the case of group SU(3). In section 4 we propose a way of obtaining Kählerian structures and Kählerian potentials on theorbits. Section5concernsastructureofthecohomologyringsoftheorbitsand findingofG-invariant basesforthecohomology groups. 2. Coadjoint OrbitsofSemisimpleLieGroups Westartwithrecallingthenotionofacoadjointorbit. LetGbeacompactsemisim- pleclassical Liegroup, gdenote thecorresponding Liealgebra, andg∗ denote the dualspacetog. LetTbethemaximaltorusofG,andhbethemaximalcommuta- tivesubalgebra (alsocalledaCartansubalgebra) ofg. Accordingly, h∗ denotesthe dualspacetoh. Definition1. Thesubset = Ad∗µ g G ofg∗iscalledacoadjointorbit µ g O { |∀ ∈ } ofGthrough µ g∗. ∈ In the case of classical Lie groups we can use the standard representations for adjointandcoadjoint operators: Ad X = gXg−1, X g, Ad∗µ = g−1µg, µ g∗. g g ∈ ∈ Comparingtheseformulasonecaneasily seethatacoadjoint orbitcoincides with anadjointone. Definethestability subgroup atapointµ g∗ asG = g G Ad∗µ=µ . The µ g ∈ { ∈ | } coadjoint operator induces a bijective correspondence between an orbit and a µ O cosetspaceG G(inthesequel, wedealwithrightcosetspaces). µ \ GeometryandTopologyofCoadjointOrbits 3 First of all, we classify the coadjoint orbits of an arbitrary semisimple group G. Obviously, each orbit isdrawnfrom aunique point, which wecall aninitial point and denote byµ . Thefollowing theorem from [1]allowsto restrict theregion of 0 searchofaninitialpoint. Theorem(R.Bott). EachorbitofthecoadjointactionofGintersectsh∗ precisely inanorbitoftheWeylgroup. Inotherwords,eachorbitisassigned toafinitenon-empty subsetofh∗. Formore detailrecallthenotionoftheWeylgroup. LetN(H)bethenormalizer ofasubset H Gin G, that isN(H) = g H g−1Hg = H . LetC(H)bethe centralizer ⊂ { ∈ | } ofH,thatisC(H)= g G g−1hg=h, h H . Obviously,C(T)= T,where { ∈ | ∀ ∈ } TisthemaximaltorusofG. Definition2. TheWeylgroupofGisthefactor-group ofN(T)overC(T): W(G)= N(T)/C(T). The Weyl group W(G) acts transitively on h∗. The action of W(G) is performed by thecoadjoint operator. It iseasy to show that W(G) is isomorphic tothe finite group generated by reflections w across the hyperplanes orthogonal to simple α rootsα: w (µ) = µ 2hµ,αi α, µ h∗, α − hα,αi ∈ where , denotesabilinear formong∗. h· ·i Definition3. Theopendomain C = µ h∗ µ,α > 0, α ∆+ { ∈ |h i ∀ ∈ } iscalledthepositiveWeylchamber. Here∆+ denotes thesetofpositiveroots. WecallthesetΓ = µ h∗ µ,α = 0 awalloftheWeylchamber. α { ∈ |h i } If we reflect the closure C of the positive Weyl chamber by elements of the Weyl groupwecoverh∗ overall: h∗ = w C. · w∈[W(G) AnorbitoftheWeylgroupW(G)isobtainedbytheactionofW(G)onapointof C. Inthecaseofgroup SU(3), twopossible types oforbits oftheWeylgroup are shown on the root diagram (see figure 1). Black points denote intersections of a coadjointorbitwithh∗andformanorbitofW(SU(3)). ThepositiveWeylchamber is filled with grey color. It has two walls: Γ and Γ ; they are the hyperplanes α1 α2 denoted by w and w . At the left, one can see a generic case, when an orbit α1 α2 of W(SU(3)) has 6 elements. It happens if an initial point lies in the interior of the positive Weyl chamber. At the right, there is a degenerate (non-generic) case, 4 JuliaBernatska†‡,PetroHolod†‡ generic orbit positive degenerate orbit w Weyl w a a 1 chamber 1 m m a a 0 0 2 3 wa a2 a3 wa 2 2 a a 1 1 w w a3 a3 Figure1. RootdiagramforSU(3). when an orbit of W(SU(3)) has 3 elements. It happens if an initial point belongs toawallofthepositiveWeylchamber. In the both cases the closed positive Weyl chamber contains a unique point of an orbitofW(G). Weobtainthefollowing Proposition 1. Eachorbit ofGisuniquely defined byaninitial pointµ h∗, 0 O ∈ whichislocatedintheclosedpositiveWeylchamberC. Ifµ liesintheinteriorof 0 thepositiveWeylchamber: µ C,itgivesrisetoagenericorbit. Ifµ belongsto 0 0 ∈ awallofthepositiveWeylchamber: µ Γ ,α ∆+,itgivesrisetoadegenerate 0 α ∈ ∈ orbit. Asmentioned above, onecandefinetheorbit through aninitial point µ h∗ Oµ0 0∈ by =G G. Note, that a stability subgroup G as µ h∗ generically coin- Oµ0 µ0\ µ ∈ cides withthemaximaltorus T. However, ifµbelongs toadegenerate orbit, then G isalagersubgroup ofGcontaining T. Therefore, wedefineagenericorbitby µ = T G, Oµ0 \ andadegenerate oneby = G G, Oµ0 µ0\ whereG =T,G T. µ0 6 µ0 ⊃ Animportant topological property ofthecoadjoint orbits isthefollowing. Almost eachorbitcanberegarded asafibrebundle overanorbitwithanorbitasafibre, exceptforthemaximaldegenerate orbits. Indeed, ifthereexistsaninitialpointµ 0 suchthatG T,onecanformacosetspaceT G . Thus,theorbit = T G µ0⊃ \ µ0 Oµ0 \ isafibrebundleoverthebaseG GwiththefibreT G : µ0\ \ µ0 = (G G,T G ,π), Oµ0 E µ0\ \ µ0 where π denotes aprojection from the orbit onto the base. Moreover, G Gand µ0\ T G arecoadjoint orbitsthemselves. Weclaimthisby µ0 \ GeometryandTopologyofCoadjointOrbits 5 Proposition 2. Suppose = G Gisnotthemaximaldegenerate orbitofG. Oµ0 µ0\ ThenasubgroupK suchthatG K G exists,and isafibrebundleover ⊃ ⊃ µ0 Oµ0 thebaseK GwiththefibreG K: \ µ0\ = (K G,G K,π). Oµ0 E \ µ0\ Weillustratetheproposition byexamples. Example1. ThegroupSU(2)hastheonlytypeoforbits: SU(2) SU(2) = CP1. O U(1) ≃ ThegroupSU(3)hasgeneric anddegenerate orbits: SU(3) SU(3) SU(3) = , SU(3) = CP2. O U(1) U(1) Od SU(2) U(1) ≃ × × Comparing the above coset spaces we see that a generic orbit SU(3) is a fibre O bundleoveradegenerate orbit SU(3) withafibre SU(2): Od O SU(3) = ( SU(3), SU(2),π) = (CP2,CP1,π). O E Od O E The group SU(4) has several types of degenerate orbits. There is a list of all possible typesoforbits: SU(4) SU(4) SU(4) = , SU(4) = , O U(1) U(1) U(1) Od1 SU(2) U(1) U(1) × × × × SU(4) SU(4) SU(4) = , SU(4) = CP3. Od2 S(U(2) U(2)) Od3 SU(3) U(1) ≃ × × As a result, there exist several representations of a generic orbit SU(4) as a fibre O bundle. Forexample, SU(4) = ( SU(4), SU(3),π) = (CP3, SU(3),π) O E Od3 O E O SU(4) = ( SU(4), SU(2),π) = ( SU(4),CP1,π). O E Od2 O E Od2 Example2. InthepaperweconsidercompactclassicalLiegroups. Theydescribe linear transformations of real, complex, and quaternionic spaces. Respectively, these groups are SO(n) over the real field, SU(n) over the complex field, and Sp(n)overthequaternionicring. Herewelistthemaximaltoriofallthesegroups, andtheirrepresentations asfibrebundles. n−1 Themaximal torus ofSU(n)isT = U(1) U(1) U(1);thegeneric type oforbitscanberepresented as z × }|×···× { SU(n) = (CPn−1, SU(n−1),π). O E O 6 JuliaBernatska†‡,PetroHolod†‡ Themaximaltorus ofSO(n)asn = 2mandn = 2m+1hasthefollowing form T = SO(2) SO(2) SO(2);thegenerictypeoforbitscanberepresented × ×···× m as | {z } SO(2m) = (G , SO(2m−2),π) 2n;2 O E O SO(2m+1) = (G , SO(2m−1),π), 2n−1;2 O E O whereG ,G denoterealGrassmanmanifolds. 2m;2 2m−1;2 n−1 The maximal torus of Sp(n) is T = U(1) U(1) U(1); the generic type oforbitscanberepresented as z × }|×···× { Sp(n) = (HPn−1, Sp(n−1),π), O E O whereHdenotes thequaternionic ring. 3. Complex Parameterization ofCoadjoint Orbits In the theory of Lie groups and Lie algebras different ways of parameterization of coadjoint orbits are available. As the most prevalent we choose a generalized stereographic projection [2]. It is named so since in the case of group SU(2) it gives the well-known stereographic projection onto the complex plane, which is the only orbit of SU(2). The generalized stereographic projection is a projection fromadualspaceontoacoadjoint orbitparameterized bycomplexcoordinates. Complex coordinates are introduced by the well-known procedure that combines Iwasawa and Gauss-Bruhat decompositions. These coordinates are often called Bruhatcoordinates [3]. C We start with complexifying a group G in the usual way: G = exp g+ig . A C { } genericorbitofGisdefinedinG byMontgomery’s diffeomorphism: C = T G P G , (1) O \ ≃ \ C wherePdenotes theminimalparabolic subgroup ofG . C Equation (1) becomes apparent from the Iwasawa decomposition G = NAK, C whereA exp ih isthereal abelian subgroup ofG ,Nisanilpotent subgroup C ≃ { } C of G , and K is the maximal compact subgroup of G . Since we consider only C compact groups G, K coincides with G. Then the Iwasawa decomposition of G hasthefollowingform C G = NAG. ItiseasytoexpressAandNintermsofrootvectors. Let∆+ bethesetofpositive roots α ofGC. ByX , X , α ∆+, denote positive and negative root vectors, α −α ∈ respectively. By H , α ∆+, denote the corresponding Cartan vectors, which α ∈ GeometryandTopologyofCoadjointOrbits 7 form a basis for the Cartan subalgebra h. According to [4], we choose X and α X sothatX X , i(X +X ) g.Then −α α −α α −α − ∈ N exp n X , n C, A exp a iH , a R. α α α α α α ≃ nα∈P∆+ o ∈ ≃ nα∈P∆+ o ∈ Inthisnotation P = NAT. Thismakes(1)evident. Inthecaseofadegenerate orbit,wehavethefollowingdiffeomorphism: C = G G P G , (2) Oµ0 µ0\ ≃ µ0\ whereG isthestabilitysubgroupandP istheparabolicsubgroupwithrespect µ0 µ0 to . ThenP = NAG ,thatproves(2). Oµ0 µ0 µ0 Ontheotherhand,GadmitsaGaussdecomposition(forthegenerictypeoforbits): C C G = NT Z, C C C whereT isthemaximaltorusofG ,andT = ATintheabovenotation; Nand Z N∗ are nilpotent subgroups of GC normalized by TC. In terms of the root ≃ vectorsintroduced above Z = exp z X , z C. α −α α nα∈P∆+ o ∈ After [4] we call a , n , z the canonical coordinates connected with the root α α α basis H , X , X α ∆+ . Thesearecoordinates inthegroupG. α α −α { | ∈ } A comparison of the Gauss and Iwasawa decompositions implies that the orbit O isdiffeomorphic tothesubgroup manifoldZ: NAG NATZ Z. (3) O ≃ NAT ≃ NAT ≃ Diffeomorphism (3) asserts that one can parameterize the orbit in terms of the O complexcoordinates z ,α ∆+ thatarecanonical coordinates inZ. α { ∈ } However, a Gauss decomposition is local. Therefore, we use a Gauss-Bruhat de- composition instead: C G = PZw. w∈\W(G) Itgivesasystem oflocalchartsontheorbit: C = P G = Zw. (4) O \ w∈\W(G) Inthecaseofadegenerateorbit ,TistobereplacedbyG ,andPbyP . It Oµ0 µ0 µ0 issufficienttotaketheintersectionoverw W(G ) W(G)in(4). Furthermore, ∈ µ0 \ inthiscase,Zhasalessnumberofcoordinates. 8 JuliaBernatska†‡,PetroHolod†‡ Proposition 3. Eachorbit of a compact semisimple Liegroup G is locally pa- O rameterized in terms of the canonical coordinates z , α ∆+ in a nilpotent α C { ∈ } subgroup ZofG according to(4). Nowweapply theabovescheme tocompact classical Liegroups, namely SO(n), SU(n), Sp(n). The scheme consists of several steps. First we parameterize the subgroupsN,A,andthegroupGintermsof z ,α ∆+ . Secondly,wechoose α { ∈ } an initial point µ in the positive closed Weyl chamber C and generate an orbit 0 bythedressingformula Oµ0 µ = g−1µ g, g G. 0 ∈ That gives a parameterization on one of the charts covering the orbit. Finally, we extendtheparameterizationtoallotherchartsbytheactionofelementsoftheWeyl groupofG. Weconsider theschemeindetail. Step1. Beingafinitegroup, eachclassical Liegroup hasamatrixrepresentation. Let aˆ be the matrix representing an element a. An Iwasawa decomposition of zˆ Zhasthefollowingform: ∈ zˆ= nˆaˆkˆ, nˆ N, aˆ A, kˆ G. (5) ∈ ∈ ∈ Onehastosolve (5) intermsof thecomplex coordinates z that appear asentries α ofthematrixzˆ. Thefollowingtransformation of(5)makesthecomputation easier zˆzˆ∗ = nˆaˆkˆkˆ∗aˆ∗nˆ∗ = nˆaˆ2nˆ∗, wherekˆ∗denotesthehermitianconjugateofkˆ. Indeed,kˆkˆ∗ = eforallofthemen- tioned groups. This is evident, if one considers the conjugation over the complex field in the case of SU(n), and over the quaternionic ring in the case of Sp(n). If kˆ SO(n) one has kˆ∗ = kˆT, and the equality kˆkˆ∗ = e is obvious. Moreover, it ∈ caneasily bechecked thataˆaˆ∗ = aˆ2. Whennˆ andaˆareparameterized intermsof z ,thematrixkˆ(z)iscomputedbytheformula α { } kˆ(z) = aˆ−1(z)nˆ−1(z)zˆ. Here we obtain complex parameterizations of N, A, G for all classical compact groupsofsmalldimensions. Example 3. In the case of group SU(n), the corresponding complexified group is SL(n,C). The subgroup N consists of complex upper triangular matrices with ones on the diagonal, the subgroup Z consists of complex low triangular matrices with ones on the diagonal, the subgroup A contains real diagonal matrices aˆ = diag(r , r , ..., r )suchthat n r = 1. 1 2 n i=1 i Q GeometryandTopologyofCoadjointOrbits 9 Decomposition (5)foragenericorbit SU(3) getstheform O 1 z11 01 00 = 01 n11 nn32r01 rr012 00uˆ, uˆ ∈ SU(3), z3 z2 1 0 0 1 0 0 r2      whenceitfollows r2 = 1+ z 2+ z z z 2, r2 = 1+ z 2+ z 2 1 | 1| | 3− 1 2| 2 | 2| | 3| 1 1 z¯ n = (z¯ (1+ z 2) z z¯ ), n = (z¯ +z z¯ ), n = 3. 1 r2 1 | 2| − 2 3 2 r2 2 1 3 3 r2 1 2 2 Thedressing matrixuˆis 1 z¯1 z¯3−z¯1z¯2 r1 −r1 − r1 uˆ = z1(1+|z2|2)−z3z¯2 1+|z3|2−z1z2z¯3 z¯2+z1z¯3. r1r2 r1r2 − r1r2  z3 z2 1   r2 r2 r2  SU(3) The case of a degenerate orbit is derived from the above by assigning Od z =0,orz =0. 1 2 Example4. Inthecaseofgroup Sp(n), thecomplexified group isSp(n,C). The both groups describe linear transformations of the quaternionic vector space Hn. Therefore, it is suitable to operate with quaternions instead of complex numbers. Eachquaternion q isdetermined bytwocomplexnumbers z ,z asq = z +z j. 1 2 1 2 The quaternionic conjugate of q is q¯ = z¯ jz¯ , where z¯ , z¯ are the complex 1 2 1 2 − conjugates ofz ,z . Severalusefulrelations areavailable: 1 2 jz =z¯j, z+w = z¯+w¯, z w = w¯ z¯, · · wherez, w C. ∈ Thesubgroups N, Zhave the samerepresentatives asin the case ofgroup SU(n), but over the quaternionic ring. Thesubgroup Aconsists of real diagonal matrices withthesamepropertyasinthecaseofSU(n). WestartwiththesimplestgroupSp(2). Supposev, q Hsuchthatv = n +n j, 1 2 ∈ q = z +z j, where n , n , z , z C. Decomposition (5) for an orbit Sp(2) 1 2 1 2 1 2 ∈ O getsthefollowingform 1 1 0 1 v 0 = r pˆ, pˆ Sp(2), (cid:18)q 1(cid:19) (cid:18)0 1(cid:19)(cid:18)0 r(cid:19) ∈ whenceitfollowsr2 = 1+ q 2,v = q¯/r2,orintermsofcomplexcoordinates: | | z¯ z r2 = z 2+ z 2, n = 1, n = 2. | 1| | 2| 1 r2 2 −r2 10 JuliaBernatska†‡,PetroHolod†‡ Thedressing matrixpˆis 1 1 z¯ +jz¯ pˆ= − 1 2 . z1 2+ z2 2 (cid:18)z1+z2j 1 (cid:19) | | | | p In the case of group Sp(3), we perform all computations in terms of quaternions. Supposeq = z +z j,q = z +z j,q = z +z j,v = n +n j,v = n +n j, 1 1 2 2 3 4 3 5 6 1 1 2 2 3 4 v = n +n j. Then,foragenericorbit Sp(3),oneobtains 3 5 6 O 1 q11 01 00 = 01 v11 vv32r01 rr012 00pˆ, pˆ∈ Sp(3), q3 q2 1 0 0 1 0 0 r2      whenceitfollows r2 =1+ q 2+ q q q 2, r2 = 1+ q 2+ q 2 1 | 1| | 3− 2 1| 2 | 2| | 3| 1 1 q¯ v = (q¯ (1+ q 2) q¯ q ), v = (q¯ +q q¯ ), v = 3. 1 r2 1 | 2| − 3 2 2 r2 2 1 3 3 r2 1 2 2 Thedressing matrixpˆis 1 q¯1 q¯3−q¯1q¯2 r1 −r1 − r1 pˆ= q1(1+|q2|2)−q¯2q3 1+|q3|2−q1q¯3q2 q¯2+q1q¯3.  r1q3r2 r1q2r2 − r11r2   r2 r2 r2  The case of Sp(n) in terms of quaternions is very similar to the case of SU(n). Theonlywarningisthatthemultiplication ofquaternions isnotcommutative. Example5. InthecaseofgroupSO(n),thecorresponding complexified groupis SO(n,C). Representatives of the subgroups N and Z have not so clear structure as for groups SU(n) and Sp(n). The real abelian subgroup A consists of block- diagonal matrices aˆ = diag(A , A , ..., A )inthecaseofgroup SO(2m), and 1 2 m aˆ = diag(A , A , ..., A , 1)inthecaseofgroupSO(2m+1). Here 1 2 m cosha isinha i i Ai = (cid:18)isinhai −coshai (cid:19). Consider thegroup SO(3). Theonly type of orbits is SO(3)=SO(2) SO(3). In O \ thiscasedecomposition (5)getstheform 1 z2 iz2 z 1 n2 in2 n cosha isinha 0 − 2 − 2 − − 2 2 −  iz2 1+ z2 iz =  in2 1+ n2 inisinha cosha 0oˆ, − 2 2 − 2 2 −  z iz 1   n in 1  0 0 1    −  