Dual non-abelian duality and the Drinfeld double

The standard notion of the non-abelian duality in string theory is generalized to the class of \sigma-models admitting `non-commutative conserved charges'. Such \sigma-models can be associated with every Lie bialgebra ({\cal G},\tilde{\cal G}) and they posses an isometry group iff the commutant [\tilde{\cal G},\tilde{\cal G}] is not equal to \tilde{\cal G}. Within the enlarged class of the backgrounds the non-abelian duality {\it is} a duality transformation in the proper sense of the word. It exchanges the roles of {\cal G} and \tilde{\cal G} 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 ({\cal G}, \tilde{\cal 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.