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

TL;DR

This paper establishes an explicit operator realization of quantum group co-product actions in 2D gravity, revealing a new type of covariance involving worldsheet variables and a central extension of $U_q(sl(2))$.

Contribution

It introduces a novel operator realization of quantum group co-product actions, including a new central extension and its relation to 2D gravity screening charges.

Findings

Derived quantum group covariance of fusion and braiding matrices.

Identified a new central extension of $U_q(sl(2))$ involving worldsheet variables.

Connected the algebra of field transformations with Virasoro descendants.

Abstract

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.