Some properties of almost abelian Lie algebras (Q2191890)

This work is devoted to the study of almost abelian Lie algebras, i.e. Lie algebras admitting an abelian ideal of codimension one. Some properties of this family of Lie algebras, denoted by \(\mathrm{AA}_n(K)\) is analyzed, where \(\mathrm K\) is any field of characteristic \(0\). There is a special subspace of \(\mathrm{AA}_n(K)\) consisting of nilpotent Lie algebras and denoted by \(\mathrm{AAN}_n(K)\). The purpose of this work is study these sets. First, a classification of almost abelian nilpotent Lie algebras is given, up to isomorphism (Theorem 1). This classes of isomorphic Lie algebras are parametrized by Young diagrams associated with the nilpotent action on the abelian ideal. Then the automorphism groups of almost abelian Lie algebras is computed. To do that, the author proves that for the class \(\mathrm{AA}_n(K)\) the notion of nonreducibility and of indecomposability are equivalent. He also show in Theorem 4 an interesting decomposition of the automorphism group of an arbitrary nonreducible Lie algebra from \(\mathrm{AA}_n(K)\). Finally, he considers a connection of almost abelian nilpotent Lie algebras with the study of irreducible components of the space of all \(n\)-dimensional Lie algebras. In particular, it is shown that the set \(\mathrm{AAN}_n(K)\) is strictly larger than the set of Lie algebras in the intersection of all irreducible components of the space of all Lie algebras.