For strongly coupled 2D-gravity, with central charge C grav = 1 +6( s + 2), s = 0, ± 1 a chiral (2, 2)-operator Φ satisfies a closed exchange algebra on the unit circle with a consistent restriction to a unitary subspace of the Virasoro representation. In this paper, this result of Gervais and Neveu is first extended to a larger unitary space, with characters equal to those of a free boson compactified on a circle with radius √2(2 − s). Second, Φ is shown to take a simple form when expressed in terms of the operators whose exchange algebra coincides with the universal R-matrix of the quantum group SL(2) q . Third, in tthe case of strongly coupled A 2 Toda theories, i.e. “W 3-extended 2D-gravity”, the generalization of the Φ-field is obtained for C T = 2 + 24( s + 2). Going from A 1 (Liouville ≡ 2D-gravity) to A 2 brings in interesting novel features, such as an intriguing U(1) gauge field configuration defined on the A 2 weight lattice.