Geometric approaches to Lie bialgebras, their classification, and applications

TL;DR

This thesis develops geometric and algebraic methods, including the Darboux family, to classify coboundary Lie bialgebras and explores their applications in Lie-Hamilton systems and deformations.

Contribution

It introduces the Darboux family for classifying r-matrices and applies algebraic and geometric techniques to classify low-dimensional coboundary Lie bialgebras.

Findings

Classified 3D coboundary Lie bialgebras using algebraic methods.

Introduced the Darboux family for studying equivalence classes of r-matrices.

Explored applications in Lie-Hamilton systems and deformations of Lie systems.

Abstract

This PhD Thesis consists of two parts. The first part focuses on novel algebraic and geometric approaches to the classification problem of coboundary Lie bialgebras up to Lie algebra automorphisms. More specifically, Grassmann, graded algebra and algebraic invariant techniques are discussed. Using these algebraic methods, equivalence classes of r-matrices for three-dimensional coboundary Lie bialgebras are studied. Moreover, particular higher-dimensional cases, e.g. $\mathfrak{so}(2,2)$ and $\mathfrak{so}(3,2)$, are partially analysed. From the geometric perspective, the main role is played by the newly introduced notion: the Darboux family. This powerful tool allows an efficient and thorough study of equivalence classes of r-matrices for four-dimensional indecomposable coboundary Lie bialgebras. In order to showcase its ability to tackle decomposable examples, $\mathfrak{gl}_2$ is additionally studied. The second part of the Thesis sketches interesting directions for applications of r-matrices. Firstly, it is illustrated how r-matrices might be useful to describe foliated Lie-Hamilton systems. Secondly, the role of r-matrices in deformations of certain cases of Lie systems is discussed. In particular, based on the general procedure for deformations of Lie-Hamilton systems, its extension to Jacobi-Lie systems is suggested and supported by the detailed computation of the deformed Schwarz equation.