AUTOISOMETRIES AND AUTOSIMILARITIES OF LIE ALGEBRA 𝒜(1) ⊕ ℛ2

We consider four-dimensional Lie algebra 𝒜(1) ⊕ ℛ2 endowed with Lorentzian scalar product. We find all the one-parameter groups of isometries and similarities, which are simultaneously automorphisms of Lie algebra, and also we find the conditions of existence of such one-parameter group. Conditions of existence are associated with the location of ideals with respect to isotropic cone.

Asymptotical Behaviour of Roots of Infinite Coxeter Groups

Let W be an infinite Coxeter group. We initiate the study of the set E of limit points of “normalized” roots (representing the directions of the roots) of W. We show that E is contained in the isotropic cone Q of the bilinear form B associated with a geometric representation, and we illustrate this property with numerous examples and pictures in rank 3 and 4. We also define a natural geometric action of W on E, and then we exhibit a countable subset of E, formed by limit points for the dihedral reflection subgroups of W. We explain how this subset is built fromthe intersection with Q of the lines passing through two positive roots, and finally we establish that it is dense in E.

Asymptotical behaviour of roots of infinite Coxeter groups I

International audience Let $W$ be an infinite Coxeter group, and $\Phi$ be the root system constructed from its geometric representation. We study the set $E$ of limit points of "normalized'' roots (representing the directions of the roots). We show that $E$ is contained in the isotropic cone $Q$ of the bilinear form associated to $W$, and illustrate this property with numerous examples and pictures in rank $3$ and $4$. We also define a natural geometric action of $W$ on $E$, for which $E$ is stable. Then we exhibit a countable subset $E_2$ of $E$, formed by limit points for the dihedral reflection subgroups of $W$; we explain how $E_2$ can be built from the intersection with $Q$ of the lines passing through two roots, and we establish that $E_2$ is dense in $E$. Soit $W$ un groupe de Coxeter infini, et $\Phi$ le système de racines construit à partir de sa représentation géométrique. Nous étudions l'ensemble $E$ des points d'accumulation des racines "normalisées'' (représentant les directions des racines). Nous montrons que $E$ est inclus dans le cône isotrope $Q$ de la forme bilinéaire associée à $W$, et nous illustrons cette propriété à l'aide de nombreux exemples et images en rang $3$ et $4$. Nous définissons une action géométrique naturelle de $W$ sur $E$, pour laquelle $E$ est stable. Puis nous présentons un sous-ensemble dénombrable $E_2$ de $E$, constitué des points d'accumulation associés aux sous-groupes de réflexion diédraux de $W$ ; nous expliquons comment $E$ peut être construit à partir des points d'intersection de $Q$ avec les droites passant par deux racines, et nous montrons que $E_2$ est dense dans $E$.