Nilpotency degree of the nilradical of a solvable Lie algebra on two generators

FERNANDO SZECHTMAN; FERNANDO LEVSTEIN; LEANDRO CAGLIERO

arXiv.org

Cornell University

Año: 2020

2331-8422

Given a sequence d⃗ =(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 d⃗ , 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.