The standard notion of the non-Abelian duality in string theory is generalized to the class of σ-models admitting 'non-commutative conserved charges'. Such σ-models can be associated with every Lie bialgebra (G,G) and they possess an isometry group iff the commutant [G,G] is not equal toG. Within the enlarged class of the backgrounds the non-Abelian duality is a duality transformation in the proper sense of the word. It exchanges the roles of G andG and it can be interpreted as a symplectomorphism of the phase spaces of the mutually dual theories. We give explicit formulas for the non-Abelian duality transformation for any (G,G). The non-Abelian analogue of the Abelian modular space O(d, d; Z) consists of all maximally isotropic decompositions of the corresponding Drinfeld double.

We point out the existence of nonlinear σ-models on group manifolds which are left symmetric and right Poisson-Lie symmetric. We discuss the corresponding rich T-duality story with particular emphasis on two examples : the anisotropic principal chiral model and the SL(2,C)/SU(2) WZW model. The latter has the de Sitter space as its (conformal) non-Abelian dual.

We prove the integrability of the Yang-Baxter σ-model which is the Poisson-Lie deformation of the principal chiral model. We find also an explicit oneto-one map transforming every solution of the principal chiral model into a solution of the deformed model. With the help of this map, the standard procedure of the dressing of the principal chiral solutions can be directly transferred into the deformed Yang-Baxter context.

A duality invariant first order action is constructed on the loop group of a Drinfeld double. It gives at the same time the description of both of the pair of σ-models related by Poisson-Lie T-duality. Remarkably, the action contains a WZW-term on the Drinfeld double not only for conformally invariant σ-models. The resulting actions of the models from the dual pair differ just by a total derivative corresponding to an ambiguity in specifying a two-form whose exterior derivative is the WZW three-form. This total derivative is nothing but the Semenov-Tian-Shansky symplectic form on the Drinfeld double and it gives directly a generating function of the canonical transformation relating the σ-models from the dual pair.

We describe the self-interacting scalar eld on the truncated sphere and we perform the quantization using the functional (path) integral approach. The theory posseses a full symmetry with respect to the isometries of the sphere. We explicitely show that the model is nite and the UV-regularization automatically takes place. CERN-TH/95-138