Closed G2-Structures on Solvable Lie Algebras with Trivial Centre

The purpose of this thesis is twofold: on the one hand, to provide the reader with a state-of-the art review of the theory on geometrical structures such as SL_3(C), SU(3) and G2-structures, and on the other hand, to tackle the currently open problem of classifying, up to isomorphisms, the closed G2-structures defined on 7-dimensional solvable Lie algebras with trivial centre. Beginning with a detailed overview of Lie algebra fundamentals—with special attention to the structure and classification of solvable Lie algebras—we then shift our attention to the study of the geometrical structures mentioned before in three different different-yet-strongly-related contexts: vector spaces, (certain) manifolds, and Lie algebras. By studying the relationships found between these structures, we present the up-to-date results in the classification of G2-structures on Lie algebras and, hopefully, shed some light on the aforementioned open problem.