A family of Poisson non-compact symmetric spaces

arXiv:math/0607102v1[math.DG]5Jul2006AFAMILYOFPOISSONNON-COMPACTSYMMETRICSPACESNICOL`SANDRUSKIEWITSCHANDALEJANDROTIRABOSCHIAbstrat.WestudyPoissonsymmetrispaesofgrouptypewithCartansubalgebraadaptedtotheLieobraket.IntrodutionLetAbeaPoisson-LiegroupandHaLiesubgroupofA.ThehomogeneousspaeA/HendowedwithaPoissonstrutureisaPoissonhomogeneousspaeiftheationA×A/H→A/HisamorphismofPoissonmanifolds.Poissonhomogeneousspaes,aftertheseminalpaper[D3℄,havebeenstudiedbyseveralauthors,see[EL,FL,L1,KRR,K,KoS℄;andalsoinonnetionwiththequantumdynamialYang-Baxterequation,see[EE,EEM,KS,L2℄andreferenestherein.IfA/HarriesaPoissonstruturesuhthenaturalprojetionA→A/HisamorphismofPoissonmanifolds,thenA/HisaPoissonhomogeneousspae,andinthisaseissaidtobeofgrouptype.AssumethatAandHareonneted.Leta,hdenotetheLiealgebrasofA,Handletδ:a→a⊗abetheLieobraketinheritedfromthePoissonstrutureofA[D1℄,seealso[KoS,Th.3.3.1℄.Thenthefollowingonditionsareequivalentsee[S,KRR℄:(i)A/HisaPoissonhomogeneousspaeofgrouptype;(ii)h⊥isasubalgebraina∗;(iii)hisaoidealofa,i.e.δ(h)⊂h⊗a+a⊗h.Inthispaper,westudyPoissonnon-ompatsymmetrispaesofgrouptype.Thatis,weassumethefollowingsetting:•A=G0isanon-ompatabsolutelysimplerealLiegroupwithniteenterandLiealgebrag0;wexaCartandeompositiong0=k0⊕p0;•thePoisson-LiegroupstrutureonG0orrespondstoanalmostfatorizableLiebialgebrastrutureong0;•H=K0isaonnetedLiesubgroupwithLiealgebrak0(inotherwords,K0isamaximalompatsubgroupofG0).WenotethatthesymmetrispaeG0/K0alwayshasastrutureofPoissonhomogeneousspae,seesubsetion1.5.However,whetherthisPoissonhomogeneousstrutureisofgrouptypeisnotevident.Date:February2,2008.1991MathematisSubjetClassiation.Primary:17B62.Seondary:53D17.ThisworkwassupportedbyCONICET,Fund.Antorhas,Ag.CrdobaCienia,FONCyTandSeyt(UNC).12ANDRUSKIEWITSCHANDTIRABOSCHIAlmostfatorizableLiebialgebrastruturesong0werelassiedin[AJ℄,startingfromtheanalogouslassiationintheomplexase[BD℄.Inpartiular,toeahalmostfatorizableLiebialgebrastrutureδong0orrespondsauniqueCartansubalgebrahoftheomplexiationgofg0,auniquesystemofsimplerootsΔinthesetofrootsΦ(g,h)andauniqueBD-tripleseesubsetion1.4.Ontheotherhand,allmaximalompatLiesubgroupsofG0areonjugated,andtheyatuallyariseasthexedpointsetoftheChevalleyinvolutionorrespondingtosomeCartansubalgebraofgandsomesystemofsimpleroots.WesaythatK0isadaptedtoδifK0isthexedpointsetoftheChevalleyinvolutionorrespondingtotheCartansubalgebrahandthesystemofsimplerootsΔdeterminedbyδ.Hereisthemainresultofthepaper.Theorem1.Let(g0,δ)beanalmostfatorizableabsolutelysimplerealLiebialgebra,letσbethesesquilinearinvolutionofgsuhthatg0=gσ,andletK0bethemaximalompatLiesubgroupofG0adaptedtoδ.•Assumethatσisoftheformς,ςμorωJ.ThenG0/K0isaPoissonhomogeneousspaeofgrouptypeifandonlyiftheBD-tripleistrivialand(g0,δ)isasinTable1.g0σTypeContinuousparameterRemarksgRςallλα,β=0su(n,n+1)A2nsu(n+1,n+1)ςμ,A2n+1λα,β=λμ(α),μ(β),so(n−1,n+1)μ6=idDnRe(λα,β+λα,μ(β))=0EIIE6Paintedroots:su(j,n+1−j)ωJAnjthrootso(2,2n−1)Bnλα,β∈iRrstrootsp(n,R)Cnnthrootso(2,2n−2)Dnrstrootso∗(2n)DnnthrootEIIIE6extremeofthelongbranhEVIIE7extremeofthelongbranhExplanationofthetable.σtheinvolutiondenedbyg0,asin(1.7),(1.8).ThepaintedrootsarelassiersofVoganlassiation.ExplanationofVoganlassiationannotationsarein[Kn℄.Table1.G0/K0Poissonhomogeneousspaeofgrouptype,K0adaptedAFAMILYOFPOISSONNON-COMPACTSYMMETRICSPACES3•Assumethatσisoftheformωμ,Jwithμ6=idandthattheBD-tripleistrivial.ThenG0/K0isaPoissonhomogeneousspaeofgrouptypeifandonlyifg0=sl(3,R)andλα,β=−λμ(α),μ(β).TheproofoftheTheoremfollowsfromPropositions2.4(forσ=ς,Γ1=Γ2=∅),2.5(forσ=ςμ,Γ1=Γ2=∅),2.6(forσ=ςorςμ,Γ16=∅),2.7(forσ=ωJ)and2.12(forσ=ωμ,J,Γ1=Γ2=∅),inpreseneoftheinformationin[AJ,Tables1.1and2.1℄summarizedinProposition1.4.Theonlyasethatremainsopeniswhenσ=ωμ,J,μ6=id,andtheBD-triplenon-trivial.Thepaperisorganizedasfollows.Setion1isdevotedtopreliminariesonLiebialgebras,inludingtheelebratedtheoremofBelavinandDrinfeld,andthelassiationresultin[AJ℄.Afterthis,weprovethemainresultinSetion2,byaase-by-aseanalysis.1.Liebialgebras1.1.SimpleLiealgebras.Inthissetionweintroduethenotationthatwillbeusedthroughallthepaper.IfθisabijetionofasetX,thenXθdenotesthexed-pointsetofθ.Ifa∈C,wedenotebyatheonjugateofa.Weseti=√−1.AlltheLiealgebrasinthispaperarenite-dimensional,unlessexpliitlystated.WedenotebygasimpleomplexLiealgebraandbyB(,):g×g→CtheKillingformong.LethbeaCartansubalgebraofg.WeextendB(,)toh∗×h∗intheusualway.WedenotebyΦ=Φ(g,h)theorrespondingrootsystem.LetΔ⊂Φbeasystemofsimpleroots.LetΦ+bethesetofpositiverootswithrespettoΔ.Givenα=Pβ∈Δnββ∈Φ+,wedenotebyℓ(α)=Pβ∈Δnβthelengthofα.Forα∈Φ,wedenehαastheelementofhthatsatisesB(hα,h)=α(h),forallhinh.Letgαbetherootspaeorrespondingtoα.Theng=g+⊕h⊕g−istheCartandeompositionofg,whereg±=⊕α∈±Φ+gα.Wehooserootvetorseα∈gα−0suhthatB(eα,e−α)=1forα∈Φ.Thentherearenon-zeroonstantsNα,βforeveryα,β∈Φ,suhthat[eα,e−α]=hα,(1.1)[eα,eβ]=Nα,βeα+β,ifα+β∈Φ,(1.2)[eα,eβ]=0,ifα+β6=0andα+β6∈Φ.(1.3)WesetNα,β=0,ifα+β6=0andα+β6∈Φ.Thus,wehaveforallα,βandγinΦ,Nα,β=−Nβ,α,(1.4)Nα,β=Nβ,γ=Nγ,α,ifα,β,γ∈Φ,α+β+γ=0.(1.5)Thefollowingfatiswell-kno