Nilpotency degree of the nilradical of a solvable Lie algebra on two generators and uniserial modules associated to free nilpotent Lie algebras

Abstract

Given a sequence d→=(d1,…,dk) of natural numbers, we consider the Lie subalgebra h of gl(d,F), where d=d1+⋯+dk and F is a field of characteristic 0, generated by two block upper triangular matrices D and E partitioned according to d→, and study the problem of computing the nilpotency degree m of the nilradical n of h. We obtain a complete answer when D and E belong to a certain family of matrices that arises naturally when attempting to classify the indecomposable modules of certain solvable Lie algebras.Our determination of m depends in an essential manner on the symmetry of E with respect to an outer automorphism of sl(d). The proof that m depends solely on this symmetry is long and delicate.As a direct application of our investigations on h and n we give a full classification of all uniserial modules of an extension of the free ℓ-step nilpotent Lie algebra on n generators when F is algebraically closed.