
Journal of Lie Theory 32 (2022), No. 3, 643670 Copyright Heldermann Verlag 2022 The Cortex of Nilpotent Lie Algebras of Dimensions Less or Equal to 7 and SemiDirect Product of Vector Groups: Nilpotent Case Béchir Dali Dept. of Mathematics, Faculty of Sciences of Bizerte, University of Carthage, Bizerte, Tunisia bechir.dali@fsb.ucarthage.tn Chaima Sayari Dept. of Mathematics, Faculty of Sciences of Bizerte, University of Carthage, Bizerte, Tunisia sayari.chayma@gmail.com [Abstractpdf] The paper deals with the cortex of real nilpotent Lie algebras. We first show that for any real nilpotent Lie algebra $\mathfrak g$ of dimension less or equal to $6$, its cortex coincides with the set of the common zeros of the $G$invariant polynomials on $\mathfrak g^\star$ namely the Icortex, where $G$ is the corresponding connected and simply connected Lie group and $\mathfrak g^\star$ is its dual. Next we give an example of $7$dimensional (real) nilpotent Lie algebra for which the cortex is a proper semialgebraic set in the Icortex. Finally we study the cortex of a class of nilpotent Lie groups given by a semidirect product of abelian groups $G:=\mathbb R^m\rtimes_\pi V$ where $\pi$ is the continuous representation of $\mathbb R^n$ on the $m$dimensional (real) vector space $V$ defined by $$ \pi(t_1,\dots,t_n)=\exp{\left(\sum_{i=1}^nt_iA_i\right)} $$ with $\{A_1,\dots, A_n\}$ is a set of pairwise commuting nilpotent matrices in $\mathbb R^{m\times m}$. Keywords: Nilpotent and solvable Lie groups, unitary representations of locally compact Lie groups. MSC: 22E25, 22E15, 22D10. [ Fulltextpdf (213 KB)] for subscribers only. 