Solutions of the Quantum-Yang-Baxter-Equation from Symmetric Spaces
We show that for each semi-Riemannian locally symmetric space the curvature tensor gives rise to a rational solution $r$ of the classical Yang-Baxter equation with spectral parameter. For several Riemannian globally symmetric spaces $M$ such as real, compl
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002 lJu 3 ]AQ.tham[ 1v8107010/tham:viXraSolutionsoftheQuantum-Yang-Baxter-Equation
fromSymmetricSpaces
MartinBordemann
LaboratoiredeMath´ematiquesetApplications
Facult´edesSciencesetTechniquesUniversit´edeHauteAlsace,Mulhouse
4,ruedesFr`eresLumi`ere68093Mulhouse,France
MartinWalter Fakult¨atf¨urPhysikUniversit¨atFreiburgHermann-Herder-Str.379104Freiburgi.Br.,Germany
FR-TH01/06July2001
Abstract
Weshowthatforeachsemi-RiemannianlocallysymmetricspacethecurvaturetensorgivesrisetoarationalsolutionroftheclassicalYang-Baxterequationwithspectralparameter.ForseveralRiemanniangloballysymmetricspacesMsuchasreal,complexandquaternionicGrassmannmanifoldsweexplicitlycomputesolutionsRofthequantumYangBaxterequations(representedinthetangentspacesofM)whichgeneralizethequantumR-matrixfoundbyZamolodchikovandZamolodchikovin1979.
1IntroductionandResults
SincetheirdiscoveriesnotonlytheclassicalYang-Baxterequation(CYBE)butalsothethequan-tumYang-Baxterequation(QYBE)areplayinganimportantr oleinseveralbranchesofphysicsandmathematics.FollowingSklyanin,theCYBEcanbeformulatedforanelementr∈g∧gonanabstractLiealgebragandtakestheform
[r12,r13]+[r12,r23]+[r13,r23]=0.
WeusethestandardnotationfromthetheoryofHopfalgebras,wheresubscriptsdescribeonwhichplacesinann-foldtensorproductthequantitiesinquestionact.Werefertosolutionsof
