arXiv subjects
Publications and source records attributed to Samuel Zamour.
We develop the theory of p-Lie algebras of finite Morley rank. In particular, we prove a form of weight spaces decomposition relative to an appropriate notion of torus and we obtain a fairly complete characterization in the soluble case.
We prove the existence of Cartan subrings, i.e., self-normalizing nilpotent subrings in soluble ranked Lie rings.
We establish vanishing results for the first cohomology group of nilpotent groups and Lie rings when the submodule of invariants is trivial. Our results are obtained within a model-theoretic setting, namely for structures that are definable in a finite-dimensional theory, which encompasses algebraic groups over algebraically closed fields, real semi-algebraic groups, and finite-dimensional Lie algebras over an algebraically or real closed field. Since classical tools - such as computations with spectral sequences and rigidity of the linear dimension - are not available in our setting, we develop an elementary algebraic approach. As applications, we derive a form of Frattini's argument for Cartan subrings and a definable version of Maschke's theorem for actions of definable connected p-divisible abelian groups, with a view toward the ongoing study of soluble finite-dimensional Lie rings.
In line with known results on groups, we show the existence of the nilpotent radical in Lie rings with minimal condition on centralizers. We also prove a form of Engel's theorem if the characteristic is zero.
We describe the classification of ranked definably quasi-Frobenius groups of odd type : dihedral configurations are isomorphic to PGL(2, K) for K an algebraically closed field of characteristic other than two; Frobenius groups are spilt and solvable if the characteristic of the underlying field is positive. To achieve this classification, we prove some results on ranked definably linear groups.
We develop the model theory of omega-stable K-loops and 'symétrons'. Continuing Poizat's seminal work, we notably establish an appropriate version of the indecomposability theorem and we adapt Lascar's analysis to this context. -- Nous développons la théorie des modèles des K-boucles et des symétrons omega-stables. En poursuivant le travail séminal de Poizat, nous établissons notamment une version appropriée du théorème des indécomposables et nous adaptons l'analyse de Lascar à ce contexte.
We are interested in a class of groups, quasi-Frobenius groups (with involutions), whose internal structure generalizes that of the classical groups GA1(C), PGL 2(C) and SO3(R) : a subgroup and its conjugates, of finite index in their normalizer and trivial mutual intersection, cover "generically" the ambient group. From the perspective of model theory, we work with the hypothesis of the existence of a good notion of dimension on definable sets (we must distinguish between the o-minimal case and the ranked case). We pay special attention to the ranked case. By studying the geometry of incidence induced by involutions, we sketch a classification of quasi-Frobenius groups and thus determine under which conditionsclassical groups can be identified in a dimensional framework -- -- Nous nous intéressons à une classe de groupes, les quasi-groupes de Frobenius (avec involutions), dont la structure interne généralise celle des groupes classiques GA1(C), PGL2(C) et SO3(R) : un sous-groupe et ses conjugués, d'indice fini dans leur normalisateur et d'intersection mutuelle triviale, recouvrent "génériquement" le groupe ambiant. Dans la perspective de la théorie des modèles, nous travaillons avec l'hypothèse de l'existence d'une bonne notion dimension sur les ensembles définissables (il faut distinguer le cas o-minimal et le cas rangé). Nous accordons une attention particulière au cas rangé. En étudiant la géométrie d'incidence induite par les involutions, nous esquissons une classification des quasi-groupes de Frobenius et nous déterminons ainsi sous quelles conditions des groupes classiques peuvent être identifiés dans un cadre dimensionnel.