Abelian subalgebras and ideals of maximal dimension in Lie algebras

Résumé

In this thesis, we have studied abelian subalgebras and ideals of Lie algebras by considering two invariants, named alpha and beta, which represent the maximum among the dimension of all the abelian subalgebras (ideals for beta) of a Lie algebra. We have developed a theoretical study in Chapter two with some general bounds and properties. After that, we have studied the cases of codimension 1, 2 and 3. We have also dealt with the obtainment of abelian subalgebras and ideals in several specific families of solvable lie algebras. Then, we have implemented an algorithmic method to compute the value of alpha and beta invariants, as well as a representative for them. Finally, some applications are shown.