On modular spaces of semisimple Drinfeld doubles

TL;DR

This paper constructs and analyzes the modular spaces of all 6-dimensional real semisimple Drinfeld doubles, revealing complex structures and implications for Poisson-Lie T-duality and T-plurality.

Contribution

It provides a comprehensive construction of modular spaces for semisimple Drinfeld doubles, highlighting their differences from Abelian cases and their decomposition into multiple non-isomorphic Manin triples.

Findings

Modular spaces include unions of homogeneous spaces of different dimensions.

Some Drinfeld doubles admit multiple non-isomorphic Manin triple decompositions.

Implications for Poisson-Lie T-duality and T-plurality are discussed.

Abstract

We construct modular spaces of all 6-dimensional real semisimple Drinfeld doubles, i.e. the sets of all possible decompositions of the Lie algebra of the Drinfeld double into Manin triples. These modular spaces are significantly different from the known one for Abelian Drinfeld double, since some of these Drinfeld doubles allow decomposition into several non-isomorphic Manin triples and their modular spaces are therefore written as unions of homogeneous spaces of different dimension. Implications for Poisson-Lie T-duality and especially Poisson-Lie T-plurality are mentioned.