Classification of standard Manin triples in dimension 4+4

Abstract

Four- and six-dimensional Drinfeld doubles were classified in the past in terms of Manin triples. We provide an important step towards the classification of eight-dimensional Drinfeld doubles by presenting an extensive list of Manin triples formed by pairs of four-dimensional Lie algebras. Due to the high complexity of the classification we focus on Manin triples formed by algebras in a certain standard form. The list contains 188 non-isomorphic Manin triples plus their duals. To apply the results, we construct several four-dimensional WZW models on non-semisimple Lie groups. Some of the WZW models are known from the literature, but new cases are presented as well. As a consequence of the construction method, the WZW models are Poisson--Lie dualizable.