Classification of standard Manin triples in dimension 4+4

TL;DR

The paper tackles the problem of classifying eight-dimensional Drinfeld doubles by systematically enumerating standard (4+4) Manin triples formed from pairs of four-dimensional Lie algebras. It adopts a two-stage strategy: first constrain candidates to standard, parameter-free algebras and their duals under permutations and scalings, then prune isomorphic cases via automorphisms and explicit MT isomorphisms, yielding $188$ non-isomorphic triples (plus duals). The results enable the construction of Poisson--Lie dualizable sigma models and, in particular, four-dimensional WZW backgrounds on non-semisimple groups, including two new models related to $H_4$ and a modified Nappi--Witten background. The work also introduces a framework for identifying Drinfeld doubles from MT inventories and discusses the physical significance of dual and plural models in string theory and supergravity contexts, with future plans to address non-standard triples. Overall, the study provides a detailed, operational catalog of standard $4+4$ MTs and demonstrates their utility in concrete Poisson--Lie constructions and WZW-type backgrounds.

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.