Operator Coproduct-Realization of Quantum Group Transformations in Two Dimensional Gravity, I.

Details Operator Coproduct-Realization of Quantum Group Transformations in Two Dimensional Gravity, I. Operator Coproduct-Realization of Quantum Group Transformations in Two Dimensional Gravity, I.

1995-03-28

arxiv.org/abs/hep-th/9503198v1

Commun.Math.Phys. 178 (1996) 147

Physics

High Energy Physics

High Energy Physics - Theory

33 pages, latex, no figures

Scientific paper

10.1007/BF02104913

A simple connection between the universal $R$ matrix of $U_q(sl(2))$ (for spins $\demi$ and $J$) and the required form of the co-product action of the Hilbert space generators of the quantum group symmetry is put forward. This gives an explicit operator realization of the co-product action on the covariant operators. It allows us to derive the quantum group covariance of the fusion and braiding matrices, although it is of a new type: the generators depend upon worldsheet variables, and obey a new central extension of $U_q(sl(2))$ realized by (what we call) fixed point commutation relations. This is explained by showing that the link between the algebra of field transformations and that of the co-product generators is much weaker than previously thought. The central charges of our extended $U_q(sl(2))$ algebra, which includes the Liouville zero-mode momentum in a nontrivial way are related to Virasoro-descendants of unity. We also show how our approach can be used to derive the Hopf algebra structure of the extended quantum-group symmetry $U_q(sl(2))\odot U_{\qhat}(sl(2))$ related to the presence of both of the screening charges of 2D gravity.

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