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Olivier Mathieu

Publications and source records attributed to Olivier Mathieu.

17 recordsLinked to original sources
arXiv2026

Filling surfaces with very few systoles

In the paper we describe hyperbolic surfaces filled by their systoles, where the total number of systoles is in $O(\frac{g}{\ln \,g})$, that is equivalent to the lower bound of Anderson, Parlier and Pittet \cite{APP}. Various papers \cite{SS}\cite{FB20}\cite{Sanki}\cite{ IM}\cite{ Mathieu} have investigated the same question, and the best previously known upper bounds where in $o(\frac{g}{{\sqrt{\ln \,g}}})$. Surprizingly the present approach is, in our opinion, much simpler than the methods of earlier papers.

math.MG↗
arXiv2026

Weakly Noetherian Lie Algebra and the Sierra-Walton Conjecture

Let K be a field of characteristic zero. Motivated by the conjecture that an enveloping algebra U(g) is Noetherian only if g is finite dimensional, we define the notion of weakly Noetherian Lie algebras. The main result, Theorem A, states that weakly Noetherian Lie algebras have a very constrained structure. In the specific case of graded Lie algebras, it implies an explicit classification of the perfect strictly weakly Noetherian Lie algebras, stated in Theorem B. The proofs of both theorems are quite long, and uses concrete results due to Tits, Formanek, Razmyslov, Grabowski and the author. The first theorem provides some insight on the desired conjecture. The second one implies the conjecture for all perfect graded Lie algebras, improving a celebrated theorem of Sierra and Walton.

math.RA↗
arXiv2025

Cuspidal modules over Superconformal algebras of rank \geq 1

According to V. Kac and J. van de Leur, the superconformal algebras are the simple $\Z$-graded Lie superalgebras of growth one which contains the Witt algebra. We describe an explicit classification of all cuspidal modules over the known supercuspidal algebras of rank $\geq 1$, and their central extensions. Our approach reveals some unnoticed phenomena. Indeed the central charge of cuspidal modules is trivial, except for one specific central extension of the contact algebra $\K(4)$. As shown in the paper, this fact also impacts the representation theory of $\K(3)$, $\CK(6)$ and $\K^{(2)}(4)$. Besides these four cases, the classification relies on general methods based on highest weight theory.

math.RT↗
arXiv2025

Free Jordan Algebras and Representations of $\widehat{\mathfrak{sl}}_2(J)$

Let $J$ be a unital Jordan algebra, and let $\widehat{\mathfrak{sl}}_2(J)$ be the universal central extension of its Tits-Kantor-Koecher Lie algebra. In Part A, we study the category of $(\widehat{\mathfrak{sl}}_2(J), SL_2(K))$-modules. We characterize the dominant $J$-spaces, which are analogous to the dominant highest weights appearing in classical settings. A family of universal envelopes $\mathcal{U}_n(J)$ associated to such modules is introduced and studied. We also prove some finiteness theorems. In Part C, we define the notion of smooth $\widehat{\mathfrak{sl}}_2(J)$-modules for augmented Jordan algebras $J$, and investigate the category of smooth modules in the spirit of Cline-Parshall-Scott highest weight categories. We show that the standard modules of this category are finite dimensional when $J$ is finitely generated. The free unital Jordan algebra $J(D)$ over $D$ variables is an elusive object, but finiteness and Ext-vanishing properties suggest that the smooth $\widehat{\mathfrak{sl}}_2(J(D))$-modules with even eigenvalues might form a generalized highest weight category. However, we prove that such an assertion would contradict recently obtained information about the growth of free Jordan algebras. See [24] and [13] for more details. It then follows that the category of smooth $\widehat{\mathfrak{sl}}_2(J(D))$-modules with even eigenvalues is not a generalized highest weight category when $D\geq 2$. Surprisingly, the proofs of most of these results make use of deep theorems of E. Zelmanov.

math.RT↗
arXiv2023

Jordan algebras and weight modules

We consider bounded weight modules for the universal central extension ${\mathfrak{sl}}_2(J)$ of the Tits-Kantor-Koecher algebra of a unital Jordan algebra $J$. Universal objects called Weyl modules are introduced and studied, and a combinatorial dominance criterion is given for analogues of highest weights. Specializing $J$ to the free Jordan algebra $J(r)$ of rank $r$, the category $\mathcal{C}^{fin}$ of finite-dimensional $\mathbb{Z}$-graded ${\mathfrak{sl}}_2(J)$-modules shares many properties with the representation theory of algebraic groups. Using a deep result of Zelmanov, we show that this subcategory admits Weyl modules. By analogy, we conjecture that $\mathcal{C}^{fin}$ is a highest weight category. The resulting homological properties would then imply cohomological vanishing results previously conjectured as a way of determining graded dimensions of free Jordan algebras.

math.RT↗
arXiv2023

Small Systle Sets and Coxeter Groups

The systoles of a hyperbolic surface Σ are the shortest closed geodesics. We say that the systoles fill the surface if the set Syst(Σ) of all systoles cuts Σ into polygons. We refine an idea of Schmutz [15] to construct closed hyperbolic surfaces Σ of arbitrarily large genus with a small set Syst(Σ) that fills. In fact, for the surfaces Σ considered, the cardinality of Syst(Σ) is in o(g/ ln g), where g is the genus of Σ. The proof is based on the theory Coxeter groups, combined with some elementary number theory.

math.GT↗
arXiv2023

Estimating the dimension of Thurston spine

For g at least 2, the Thurston spine Pg is the subspace of Teichmueller space Tg , consisting of the marked surfaces for which the set of shortest curves, the systoles, cuts the surface into polygons. Our main result is the existence of an infinite set A of integers such that codim Pg is a o(g/ log g), when g varies over A. This proves the recent conjecture of M. Fortier Bourque.

math.GT↗
arXiv2023

Linearity and Nonlinearity of Groups of Polynomial Automorphisms of the Plane

Given a field $K$, we investigate which subgroups of the group Aut$\mathbb{A}^2_K$ of polynomial automorphisms of the plane are linear or not. The results are contrasted. The group Aut$\mathbb{A}^2_K$ itself is nonlinear, except if $K$ is finite, but it contains some large "finite-codimensional" subgroups which are linear. This phenomenon is specific to dimension two: it is easy to prove that any "finite-codimensional" subgroup of Aut$\mathbb{A}^3_K$ is nonlinear, even for a finite field $K$. When ch$K = 0$, we also look at a similar questions for f.g. subgroups, and the results are again disparate. The group Aut$\mathbb{A}^2_K$ has a one-related f.g. subgroup which is not linear. However, there is a large subgroup, of "co-dimension-three", which is locally linear but not linear. This paper is respectfully dedicated to the memory of Jacques Tits.

math.GR↗
arXiv2022

Linearity and Nonlinearity of groups of polynomial automorphisms

Let $K$ be a field, and let $\Aut \,K^2$ be the group of polynomial automorphisms of $K^2$. If $K$ is infinite, this group is nonlinear. Moreover it contains nonlinear FG subgroups when $\ch\,K=0$. On the opposite, it contains some linear "finite codimension" subgroups. This phenomenon is specific to dimension two: it is also proved that "finite codimension" subgroups of $\Aut\,K^3$ are nonlinear, even for a finite field $K$.

math.GR↗
arXiv2022

Linearity and Non-linearity of Groups of Polynomial Automorphisms of $K^2$

Let $K$ be a field, and let $\Aut \,K^2$ be the group of polynomial automorphisms of $K^2$. We investigate which subgroups are linear or not. In characteristic zero, there are small nonlinear subgroups and some big linear subgroups. When $K$ has finite characteristic, the whole group $\Aut\,K^2$ is linear whenever $K$ is finite, and nonlinear otherwise.

math.GR↗
arXiv2021

Which Nilpotent Groups are Self-Similar?

Let $Γ$ be a finitely generated torsion free nilpotent group, and let $A^ω$ be the space of infinite words over a finite alphabet $A$. We investigate two types of self-similar actions of $Γ$ on $A^ω$, namely the faithfull actions with dense orbits and the free actions. A criterion for the existence of a self-similar action of each type is established. Two corollaries about the nilmanifolds are deduced. The first involves the nilmanifolds endowed with an Anosov diffeomorphism, and the second about the existence of an affine structure. Then we investigate the virtual actions of $Γ$, i.e. actions of a subgroup $Γ'$ of finite index. A formula, with some number theoretical content, is found for the minimal cardinal of an alphabet $A$ endowed with a virtual self-similar action on $A^ω$ of each type.

math.GR↗
arXiv2019

Population pharmacokinetics of levobupivacaine during a transversus abdominis plane block in children

BACKGROUND:Levobupivacaine is commonly used during transversus abdominis plane block in pediatric patients. However, the dosing regimen is still empirical, and the pharmacokinetic properties of levobupivacaine are not considered. Here, the pharmacokinetics of levobupivacaine during an ultrasound-guided transversus abdominis plane block were evaluated to optimize dosing regimen, with regard to the between-subject variability and the volume of levobupivacaine injected.METHOD:The clinical trial (prospective, randomized, double-blind study protocol) was conducted in 40 children aged 1 to 5 years, who were scheduled for inguinal surgery. Each patient received 0.4 mg/kg of levobupivacaine with a volume of local anesthesia solution adjusted to 0.2 mL/kg of 0.2% or 0.4 mL/kg of 0.1% levobupivacaine. Blood samples were collected at 5, 15, 20, 25, 30, 45, 60, and 75 min following the block injection. The population pharmacokinetic analysis was performed using the NONMEM software.RESULTS:From the pharmacokinetic parameters obtained, median Cmax, tmax, and area under the concentration versus time curve were 0.315 mg/L, 17 min, and 41 mg/L. min, respectively. Between-subject variability (BSV) of clearance was explained by weight. At the dose regimen of 0.4 mg/kg, none of the infants showed signs of toxicity, but in 13 patients, transversus abdominis plane block failed. After analysis, BSV for absorption rate constant, distribution volume, and clearance were 81%, 47%, and 41%, respectively. Residual unexplained variability was estimated to be 14%.CONCLUSION:For improved efficiency in the pediatric population, the dose of levobupivacaine should be greater than 0.4 mg/kg. Children's weight should be considered to anticipate any risk of toxicity.

stat.AP↗
arXiv2019

On the Free Jordan Algebras

A conjecture for the dimension and the character of the homogenous components of the free Jordan algebras is proposed. As a support of the conjecture, some numerical evidences are generated by a computer and some new theoretical results are proved. One of them is the cyclicity of the Jordan operad.

math.RT↗
arXiv2012

A Global version of Grozman's theorem

Let X be a manifold. The classification of all equivariant bilinear maps between tensor density modules over X has been investigated by Yu Grozman, who has provided a full classification for those which are differential operators. Here, we investigate the same question without the hypothesis that the maps are differential operators. In our paper, the geometric context is algebraic geometry and the manifold X is the circle Spec C[z,z^{-1}].

math.RT↗
arXiv2012

Classification of Simple Lie Algebras on a Lattice

Let $Λ$ be a lattice of rank $n$. A Lie algebra on the lattice $Λ$ is a Lie algebra ${\cal L}=\oplus_{λ\inΛ}\,{\cal L}_λ$ such that $\dim\,{\cal L}_λ=1$ for all $λ$. In this article, we classify all simple graded Lie algebras on a lattice.

math.RT↗
arXiv2008

On a symmetric space attached to polyzeta values

Quickly convergent series are given to compute polyzeta numbers. The formula involves an intricate combination of (generalized) polylogarithms at 1/2. However, the combinatorics has a very simple geometric interpretation: it corresponds with the square map on some symmetric space P.

math.NT↗