OPERATOR COPRODUCT-REALIZATION OF QUANTUM GROUP TRANSFORMATIONS IN TWO DIMENSIONAL GRAVITY.pdf

P o s t S c r i p t? p r o c e s s e d b y t h e S L A C / D E S Y L i b r a r i e s o n 2 8 M a r 1 9 9 5 . H E P - T H - 9 5 0 3 1 9 8 CERN-TH/95-48, LPTENS{95/11hep-th@xxx/9503198OPERATOR COPRODUCT-REALIZATIONOF QUANTUM GROUP TRANSFORMATIONSIN TWO DIMENSIONAL GRAVITY I.1Eugene CREMMERJean-Loup GERVAIS2Laboratoire de Physique Theorique de lEcole Normale Superieure3,24 rue Lhomond, 75231 Paris CEDEX 05, France.Jens SCHNITTGERCERN Theory Division, CH-1211 Geneva 23, Switzerland.AbstractA simple connection between the universal R matrix of Uq(sl(2)) (for spins12 and J) and the required form of the co-product action of the Hilbert spacegenerators of the quantum group symmetry is put forward. This leads usto an explicit operator realization of the co-product action on the covariantoperators. It allows us to derive the expected quantum group covariance ofthe fusion and braiding matrices, although it is of a new type: the gener-ators depend upon worldsheet variables, and obey a new central extensionof the Uq(sl(2)) algebra realized by (what we call) xed point commutationrelations. This is explained by showing on a general ground that the linkbetween the algebra of eld transformations and that of the co-product gen-erators is much weaker than previously thought. The central charges of ourextended Uq(sl(2)) algebra, which includes the Liouville zero-mode momen-tum in a nontrivial way, are related to Virasoro-descendants of unity. Wealso show how our approach can be used to derive the Hopf algebra structureof the extended quantum-group symmetry Uq(sl(2))  Ubq(sl(2)) related tothe presence of both of the screening charges of 2D gravity.1partially supported by the E.U. network \Capital Humain et Mobilite contracts #CHRXCT920069, and CHRXCT9200352Part time visitor at Theory Division CERN.3Unite Propre du Centre National de la Recherche Scientique, associee a lEcole Nor-male Superieure et a lUniversite de Paris-Sud. 1 IntroductionThe quantum group struc