Geometric Approaches to Lie Bialgebras, their Classification, and Applications

This thesis presents novel algebraic and geometric approaches to the classification problem of coboundary Lie bialgebras up to Lie algebra automorphisms. This entails the analysis of modified and classical Yang-Baxter equations and other related mathematical structures. More specifically, we develop Grassmann, graded algebra, and algebraic invariant techniques for the classification problem of coboundary Lie bialgebras and the so-called $r$-matrices. The devised techniques are mainly focused on the study of $r$-matrices for three-dimensional and indecomposable four-dimensional Lie algebras. An example of a decomposable four-dimensional Lie algebra, namely $\mathfrak{gl}_2$, is also considered. Other particular higher-dimensional Lie bialgebras, e.g. $\mathfrak{so}(2,2)$ and $\mathfrak{so}(3,2)$, are partially studied. Special relevance has the development of a new notion: the Darboux families, which provide a powerful method for the classification and determination of classes of $r$-matrices. In fact, the classification of $r$-matrices for four-dimensional indecomposable Lie bialgebras is a significative advance relative to previous results in the literature as the classification problem of Lie bialgebras has been predominantly treated algebraically in the literature. Meanwhile, we address the problem in a more geometric manner. Moreover, the theory of Lie bialgebras and related notions are employed in the study of Lie systems, their generalisations, and Hamiltonian systems. More specifically, we study the use of $r$-matrices for the description of interesting Hamiltonian systems relative to symplectic and Poisson structures. Additionally, we extend the construction of integrable deformations of Lie--Hamilton systems on symplectic manifolds to Jacobi manifolds. The outlook of this doctoral thesis suggests some further research directions, both in the abstract classification problem and the mathematical-physics applications. Appendices present a new practical method to obtain matrix representation for Lie algebras with nontrivial centre used throughout the work and the code written in Mathematica that has been used to facilitate some of the computations for coboundary Lie bialgebras classifications.