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

TL;DR

This paper explores the relationships between Lagrangian and orthogonal splittings in quadratic vector spaces, complex product structures, and their applications to Manin triples and quadratic Lie algebras, revealing new structural equivalences.

Contribution

It establishes a novel equivalence between Lagrangian and orthogonal splittings and complex product structures, and shows how to transform certain Manin triples into ones with orthogonal Lie ideal splittings.

Findings

Equivalence between Lagrangian and orthogonal splittings and complex product structures.

Transformation of Manin triples with generalized metrics into triples with orthogonal Lie ideal splittings.

Quadratic Lie algebra decompositions into orthogonal sums of anti-isomorphic Lie algebras.

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.