Construction of rank-one solvable rigid Lie algebras with nilradicals of a decreasing nilpotence index

Abstract

It is shown that for any integers 𝑘≥2 , 𝑞≥2𝑘 and 𝑁≥𝑘+𝑞+2 , there exists a real solvable Lie algebra of the first rank with a maximal torus of derivations 𝔱 possessing the eigenvalue spectrum spec(𝔱)=(1,2,⋯,𝑘,𝑞,𝑞+1⋯,𝑁), a nilradical of the nilpotence index 𝑁−𝑘 and a characteristic sequence (𝑁−𝑘,1𝑘). .