Tóm tắt
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 we 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. We 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.
