Journal of Lie Theory 32 (2022), No. 3, 643--670
Copyright Heldermann Verlag 2022
The Cortex of Nilpotent Lie Algebras of Dimensions Less or Equal to 7 and Semi-Direct Product of Vector Groups: Nilpotent Case
Béchir Dali
Dept. of Mathematics, Faculty of Sciences of Bizerte, University of Carthage, Bizerte, Tunisia
bechir.dali@fsb.u-carthage.tn
Chaima Sayari
Dept. of Mathematics, Faculty of Sciences of Bizerte, University of Carthage, Bizerte, Tunisia
sayari.chayma@gmail.com
[Abstract-pdf]
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 $\mathbb R^{m\times m}$.
Keywords: Nilpotent and solvable Lie groups, unitary representations of locally compact Lie groups.
MSC: 22E25, 22E15, 22D10.
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