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

MONTANI, H.

JOURNAL OF MATHEMATICAL PHYSICS

AMER INST PHYSICS

Año: 2023 vol. 64

0022-2488

We study complex product structures on quadratic vector spaces and on quadratic Lie algebras analyzing the Lagrangian and orthogonal splittingsassociated 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 anti-isomorphic Lie algebras, following similar steps as in the previous case, can be turned into a Manin triple admitting an orthogonalsplitting into Lie ideals.