Remarks on realizing classical quantum W3 symmetry

In this paper the relation between Jordan algebras and the nonlinear W3 algebra is explored quantum mechanically. Realization of classical W3 symmetry assumes the existence of some constant coefficients dijk (i,j,k = 1,...,D) obeying some algebraic constraints. Recent works produced solutions to these constraints and established a link with Jordan algebras for the four special dimensions D = 5, 8, 14 and 26. In the present work the authors consider a general free field realization of quantum W3 and show that this relation with Jordan algebras breaks down at least for D = 5 and 8. The authors also present some general solutions to the dijk constraints for D = 2 and D = 3 cases. The D = 2 solution is then used in the free field construction and Fateev and Zamolodchikov's realization is obtained as a special case of this solution