The Cortex of Nilpotent Lie Algebras of Dimensions Less or Equal to 7 and Semi-Direct Product of Vector Groups: Nilpotent Case

Béchir Dali¹ ; Chaima Sayari¹

¹Dept. of Mathematics, Faculty of Sciences of Bizerte, University of Carthage, Bizerte, Tunisia

Journal of Lie Theory, Tome 32 (2022) no. 3, pp. 643-670

Résumé

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 I-cortex, 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 semi-algebraic set in the I-cortex. Finally we study the cortex of a class of nilpotent Lie groups given by a semi-direct 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 $\mathbb R^{m\times m}$.

Détail

Classification : 22E25, 22E15, 22D10
Mots-clés : Nilpotent and solvable Lie groups, unitary representations of locally compact Lie groups