Starting from the classification of real Manin triples we look for those that are isomorphic as 6-dimensional Drinfeld doubles i.e. Lie algebras with the ad-invariant form used for construction of the Manin triples. We use several invariants of the Lie algebras to distinguish the non-isomorphic structures and give explicit form of maps between Manin triples that are decompositions of isomorphic Drinfeld doubles. The result is a complete list of 6-dimensional real Drinfeld doubles. It consists of 22 classes of non-isomorphic Drinfeld doubles.

Four-dimensional Manin triples and Drinfeld doubles are classified and corresponding two-dimensional Poisson-Lie T-dual sigma models on them are constructed. The simplest example of a Drinfeld double allowing decomposition into two nontrivially different Manin triples is presented.

A scheme suitable for describing quantum non-ultralocal models, including supersymmetric ones, is proposed. Braided algebras are generalized to be used through Baxterization for constructing braided quantum Yang—Baxter equations. Supersymmetric and some known non-ultralocal models are derived in the framework of this approach. As further applications of the scheme, the construction of new quantum integrable non-ultralocal models, like mKdV and anyonic supersymmetric models including deformed anyonic superalgebra, is outlined.

We give a classification of non-Abelian T-duals of the flat metric in D = 4 dimensions with respect to the four-dimensional continuous subgroups of the Poincaré group. After dualizing the flat background, we identify majority of dual models as conformal sigma models in plane-parallel wave backgrounds, most of them having torsion. We give their form in Brinkmann coordinates. We find, besides the planeparallel waves, several diagonalizable curved metrics with nontrivial scalar curvature and torsion. Using the non-Abelian T-duality, we find general solution of the classical field equations for all the sigma models in terms of d'Alembert solutions of the wave equation.

We give the classification of T-duals of the flat background in four dimensions with respect to one-, two-, and three-dimensional subgroups of the Poincaré group using non-Abelian T-duality with spectators. As duals we find backgrounds for sigma models in the form of plane-parallel waves or diagonalizable curved metrics often with torsion. Among others, we find exactly solvable time-dependent isotropic pp-wave, singular ppwaves, or generalized plane wave (K-model).