Geometric approaches to Lie bialgebras, their classification, and applications

TL;DR

This work develops a geometric–algebraic framework for classifying coboundary Lie bialgebras up to Lie algebra automorphisms, focusing on r-matrices for low-dimensional algebras and extending to higher-dimensional examples. It introduces invariant-theoretic tools via Grassmann algebras, Killing-type forms, and gradations, and leverages reduced spaces in the mCYBE to streamline CYBE/mCYBE analysis. The centerpiece is the Darboux-family approach, which geometrically encodes CYBE solutions and yields a systematic Darboux-tree method to classify four-dimensional indecomposable algebras (and related decomposable cases like gl_2); explicit orbit structures and representative r-matrices are obtained. The thesis also connects Lie bialgebras to applications in foliated Lie–Hamilton systems and their deformations on Jacobi manifolds, broadening the scope to Poisson–Lie group structures and Jacobi–lc.s. geometries, with extensive computational appendices and Mathematica code supporting the classifications. Collectively, these results advance the geometric understanding of bialgebra classifications and provide practical tools for constructing and deforming integrable systems in mathematical physics.

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.

Figures (18)

• Figure 1: Root decomposition of $\mathfrak{so}(2,2)$ (left), and the induced decompositions on $\Lambda^2 \mathfrak{so}(2,2)$ (center) and $\Lambda^3 \mathfrak{so}(2,2)$ (right) given by Theorem \ref{['thm:root']}. Non-zero homogeneous spaces are indicated by bold points. A basis for each homogeneous subspace of $\mathfrak{so}(2,2)$ is detailed in the first diagram. Limit homogeneous spaces are represented by points over a dashed line.
• Figure 2: Graded and induced decompositions of $\mathfrak{so}(3,2)$ (left), $\Lambda^2\mathfrak{so}(3,2)$ (center), $\Lambda^3\mathfrak{so}(3,2)$ (right) and limit homogeneous spaces (subspaces over the dashed line).
• Figure 3: Orbits of ${\rm Aut}(\mathfrak{sl}_2)$ acting on $\Lambda^2 \mathfrak{sl}_2$.
• Figure 4: An orbit of ${\rm Aut}(\mathfrak{su}_2)$ acting on $\Lambda^2 \mathfrak{su}_2$.
• Figure 5: Orbits of ${\rm Inn}(\mathfrak{h})$ acting on $\Lambda^2 \mathfrak{h}$.

Theorems & Definitions (181)

• Definition 1.1.1
• Proposition 1.1.2
• Example 1
• Example 2
• Example 3
• Theorem 1.2.1
• Example 4
• Theorem 1.2.2
• Theorem 1.2.3
• Example 5