Poisson–Lie T–plurality of three–dimensional conformally invariant sigma models

Starting from the classification of 6–dimensional real Drinfeld doubles and their decomposition into Manin triples we construct 3–dimensional Poisson–Lie T–dual or more precisely T–plural sigma models. Of special interest are those that are conformally invariant. Examples of models that satisfy vanishing β–function equations with zero dilaton are presented and their duals are calculated. It turns out that for ”traceless” dual algebras they satisfy the β–function equations as well but usually with rather nontrivial dilaton. We also present explicit examples of several kinds of obstacles and difficulties present in construction of quantum dual models. Such concrete examples might be helpful in further development and improvement of quantum version of Poisson-Lie T–duality.