k-Step Nilpotent Lie Algebras

TL;DR

This paper introduces a new approach to classify 2 and 3-step nilpotent Lie algebras using deformation cohomology, identifying rigid algebras and generating others through contractions.

Contribution

It proposes a deformation and contraction-based classification method for 2 and 3-step nilpotent Lie algebras, including the description of deformation cohomology and rigidity.

Findings

Identification of rigid nilpotent Lie algebras.

Classification of algebras via deformation and contraction.

Extension of classification to higher steps through contractions.

Abstract

The classification of complex of real finite dimensional Lie algebras which are not semi simple is still in its early stages. For example the nilpotent Lie algebras are classified only up to the dimension 7. Moreover, to recognize a given Lie algebra in a classification list is not so easy. In this work we propose a different approach to this problem. We determine families for some fixed invariants, the classification follows by a deformation process or contraction process. We focus on the case of 2 and 3-step nilpotent Lie algebras. We describe in both cases a deformation cohomology of this type of algebras and the algebras which are rigid regarding this cohomology. Other $p$-step nilpotent Lie algebras are obtained by contraction of the rigid ones.