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

Abstract

We study Lagrangian and orthogonal splittings\textbf{\ }of quadratic vector spaces establishing an equivalence with complex product structures. Then we show that a Manin triple equipped with generalized metric $\mathcal{G}+% \mathcal{B}$ such that $\mathcal{B}$ is an $\mathcal{O}$-operator with extension $\mathcal{G}$ of mass -1 can be turned in another Manin triple that admits also an orthogonal splitting in\textbf{\ }Lie ideals. Conversely, a quadratic Lie algebra orthogonal direct sum of a pair anti-isomorphic Lie algebras, after similar steps as in the previous case, can be turned in a Manin triple admitting an orthogonal splitting into Lie ideals.