Lagrangian and orthogonal splittings, quasitriangular Lie bialgebras, and almost complex product structures

Abstract

We study complex product structures on quadratic vector spaces and on quadratic Lie algebras analyzing the Lagrangian and orthogonal splittings associated with them. We show that a Manin triple equipped with generalized metric G+B such that B is an O-operator with extension G of mass −1 can be turned into another Manin triple that admits also an orthogonal splitting in Lie ideals. Conversely, a quadratic Lie algebra orthogonal direct sum of a pair of anti-isomorphic Lie algebras, following similar steps as in the previous case, can be turned into a Manin triple admitting an orthogonal splitting into Lie ideals.