lectures on hopf algebras quantum groups and twists

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1、DISTA-UPO/07LecturesonHopfAlgebras,⋆QuantumGroupsandTwistsPaoloAschieriCentroStudieRicercheEnricoFermi,CompendioViminale,I-00184,Roma,ItalyDipartimentodiScienzeeTecnologieAvanzateUniversit´adelPiemonteOrientale,andINFN-SezionediTorinoViaBellini25/G15100Alessandria,ItalyAbstractLeadbyexamplesweintrod

2、ucethenotionsofHopfalgebraandquantumgroup.WestudytheirgeometryandinparticulartheirLiealgebra(ofleftinvariantvectorfields).Theexamplesofthequantumsl(2)Liealgebraandofthequantum(twisted)Poincar´eLiealgebraisoθ(3,1)arepresented.arXiv:hep-th/0703013v12Mar2007⋆LecturesgivenatthesecondModaveSummerSchoolinM

3、athematicalPhysics,August6-12,2006.E-mail:aschieri@mfn.unipmn.it1IntroductionHopfalgebraswhereinitiallyconsideredmorethanHalfacenturyago.Newimpor-tantexamples,namedquantumgroups,wherestudiedinthe80’s[1–3],theyaroseinthestudyofthequantuminversescatteringmethodinintegrablesystems.Quantumgroupscanbesee

4、nassymmetrygroupsofnoncommutativespaces,thisisonerea-sontheyhavebeeninvestigatedinphysicsandmathematicalphysics(noncommutativespacesariseasquantizationofcommutativeones).TheemergenceofgaugetheoriesonnoncommutativespacesinopenstringtheoryinthepresenceofaNS2-formback-ground[4]hasfurthermotivatedthestu

5、dyofnoncommutativespaces,andoftheirsymmetryproperties.WehereintroducethebasicconceptsofquantumgroupandofitsLiealgebraofinfinitesimaltransformations.Wepedagogicallystresstheconnectionwiththeclassical(commutative)caseandwetreattwomainexamples,thequantumsl(2)LiealgebraandthequantumPoincar´eLiealgebra.Se

6、ction2showshowcommutativeHopfalgebrasemergefromgroups.Thequan-tumgroupSLq(2)isthenpresentedanditscorrespondinguniversalenvelopingalgebraUq(sl(2))discussed.TherelationbetweenSLq(2)andUq(sl(2))isstudiedinSection5.Thequantumsl(2)Liealgebra,i.e.thealgebraofinfinitesimaltransformations,isthenstudiedinSect

7、ion6.SimilarlythegeometryofHopfalgebrasobtainedfrom(abelian)twistsisstudiedviatheexampleofthePoincar´eLiealgebra.IntheappendixforreferencewereviewsomebasicalgebranotionsanddefineHopfalgebrasdiagramatic