arXiv subjects
Publications and source records attributed to Jean-Yves Charbonnel.
In this note, we are interested in the Jacobian Conjecture. Following the results of L.M.~Dru$\dot{\rm z}$kowski, we consider some vector fields depending on a certain étale polynomial map. From results of semialgebraic geometry with the consideration of some integral curves of these vector fields, we deduce that an étale polynomial endomorphism is a polynomial automorphism.
The commuting variety of a reductive Lie algebra $\mathfrak{g}$ is the underlying variety of a well defined subscheme of $\mathfrak{g}\times\mathfrak{g}$. In this note, it is proved that this scheme is normal and Cohen-Macaulay. In particular, its ideal of definition is a prime ideal. As a matter of fact, this theorem results from a so called Property (P) for a simple Lie algebra. This property says that some cohomology complexes are exact.
In this article, we are interested in some subspaces of the exterior algebra of a simple Lie algebra g. In particular, we prove that some graded subspaces of degree d generate the g-module d (g) for some integers d.
Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra of rank $\ell$ over an algebraically closed field $\Bbbk$ of characteristic zero, and let $(e,h,f)$ be an $\mathfrak{sl}_2$-triple of g. Denote by $\mathfrak{g}^{e}$ the centralizer of $e$ in $\mathfrak{g}$ and by ${\rm S}(\mathfrak{g}^{e})^{\mathfrak{g}^{e}}$ the algebra of symmetric invariants of $\mathfrak{g}^{e}$. We say that $e$ is good if the nullvariety of some $\ell$ homogenous elements of ${\rm S}(\mathfrak{g}^{e})^{\mathfrak{g}^{e}}$ in $(\mathfrak{g}^{e})^{*}$ has codimension $\ell$. If $e$ is good then ${\rm S}(\mathfrak{g}^{e})^{\mathfrak{g}^{e}}$ is a polynomial algebra. In this paper, we prove that the converse of the main result of arXiv:1309.6993 is true. Namely, we prove that $e$ is good if and only if for some homogenous generating sequence $q_1,\ldots,q_\ell$, the initial homogenous components of their restrictions to $e+\mathfrak{g}^{f}$ are algebraically independent over $\Bbbk$.
For a reductive Lie algbera over an algbraically closed field of charasteristic zero,we consider a borel subgroup $B$ of its adjoint group, a Cartan subalgebra contained inthe Lie algebra of $B$ and the closure $X$ of its orbit under $B$ in the Grassmannian.The variety $X$ plays an important role in the study of the commuting variety. In thisnote, we prove that $X$ is Gorenstein with rational singularities.
The nilpotent cone of a reductive Lie algebra has a desingularization given by thecotangent bundle of the flag variety. Analogously, the nullcone of a cartesianpower of the algebra has a desingularization given by a vector bundle over theflag variety. As for the nullcone, the subvariety of elements whose componentsare in a same Borel subalgebra, has a desingularization given by a vector bundle overthe flag variety. In this note, some properties of these varieties are given. Forthe study of the commuting variety, the analogous variety to the flag variety isthe closure in the Grassmannian of the set of Cartan subalgebras. So someproperties of this variety are given. In particular, it is smooth in codimension $1$.We introduce the generalized isospectral commuting varieties and give some properties.Furthermore, desingularizations of these varieties are given by fiber bundles over adesingularization of the closure in the grassmannian of the set of Cartan subalgebrascontained in a given Borel subalgebra.
For a finite dimensional complex Lie algebra, its index is the minimal dimension of stabilizers for the coadjoint action. A famous conjecture due to Elashvili says that the index of the centralizer of an element of a reductive Lie algebra is equal to the rank. That conjecture caught attention of several Lie theorists for years. In this paper we give an almost general proof of that conjecture.
Let g be a finite-dimensional simple Lie algebra of rank r over an algebraically closed field of characteristic zero, and let e be a nilpotent element of g. Denote by g^e the centralizer of e in g and by S(g^e)^{g^e} the algebra of symmetric invariants of g^e. We say that e is good if the nullvariety of some r homogeneous elements of S(g^e)^{g^e} in the dual of g^{e} has codimension r. If e is good then S(g^e)^{g^e} is polynomial. The main result of this paper stipulates that if for some homogeneous generators of S(g^e)^{g^e}, the initial homogeneous component of their restrictions to e+g^f are algebraically independent, with (e,h,f) an sl2-triple of g, then e is good. As applications, we obtain new examples of nilpotent elements that verify the above polynomiality condition, in in simple Lie algebras of both classical and exceptional types. We also give a counter-example in type D_7.
The generalized commuting and isospectral commuting varieties of a reductive Lie algebra have been introduced in a preceding article. In this note, it is proved that their normalizations are Gorenstein with rational singularities. Moreover, their canonical modules are free of rank 1. In particular, the usual commuting variety is Gorenstein with rational singularities and its canonical module is free of rank 1.
The commuting variety of a reductive Lie algebra ${\goth g}$ is the underlying variety of a well defined subscheme of $\gg g{}$. In this note, it is proved that this scheme is normal. In particular, its ideal of definition is a prime ideal.
The nilpotent bicone of a finite dimensional complex reductive Lie algebra g is the subset of elements in g x g whose subspace generated by the components is contained in the nilpotent cone of g. The main result of this note is that the nilpotent bicone is a complete intersection. This affirmatively answers a conjecture of Kraft-Wallach concerning the nullcone. In addition, we introduce and study the characteristic submodule of g. The properties of the nilpotent bicone and the characteristic submodule are known to be very important for the understanding of the commuting variety and its ideal of definition. In order to study the nilpotent bicone, we introduce another subvariety, the principal bicone. The nilpotent bicone, as well as the principal bicone, are linked to jet schemes. We study their dimensions using arguments from motivic integration. Namely, we follow methods developed in http://arxiv.org/abs/math/0008002v5 .
In this note we give an example of affine quotient $G/H$ where $G$ is an affine algebraic group over an algebraically closed field of characteristic 0 and $H$ is a unipotent subgroup not contained in the unipotent radical of $G$. Some remarks about symmetric algebras of centralizers of nilpotent elements in simple Lie algebras, in particular cases, are added.
Let ${\goth g}$ be a finite dimensional complex reductive Lie algebra and $\dv ..$ an invariant non degenerated bilinear form on ${\goth g}\times {\goth g}$ which extends the Killing form of $[{\goth g},{\goth g}]$. We define the homology complex $C_{\bullet}({\goth g})$. Its space is the algebra $\tk {\Bbb C}{\e Sg}\tk {\Bbb C}{\e Sg}\ex {}{\goth g}$ where $\e Sg$ and $\ex {}{\goth g}$ are the symmetric and exterior algebras of ${\goth g}$. The differential of $C_{\bullet}({\goth g})$ is the $\tk {\Bbb C}{\e Sg}\e Sg$-derivation which associates to the element $v$ of ${\goth g}$ the function $(x,y)\mapsto \dv v{[x,y]}$ on ${\goth g}\times {\goth g}$. Then the complex $C_{\bullet}({\goth g})$ has no homology in degree strictly bigger than $\rk {\goth g}$.
Let $g$ be a finite dimensional complex reductive Lie algebra and <.,.> an invariant non degenerated bilinear form on $g\times g$ which extends the Killing form of $[g,g]$. We define a subcomplex $E\_{\bullet}(g)$ of the canonical complex $C\_{\bullet}(g)$ of $g$. There exists a well defined sub-module $B\_{g}$ of the module of polynomial maps from $g\times g$ to $g$ which is free of rank equal to the dimension b of the borel subalgebras of $g$. Moreover, $B\_{g}$ is contained in the space of cycles of the canonical complex of $g$. The complex $E\_{\bullet}(g)$ is the ideal of $C\_{\bullet}(g)$ generated the exterior power of degree b of the module $B\_{g}$. We denote by ${\cal N}\_{g}$ the set of elements in $g\times g$ whose components generate a subsbspace contained in the nilpotent cone of $g$ and we say that $g$ has property (N) if the codimension of ${\cal N}\_{g}$ in $g\times g$ is strictly bigger than the dimension of the space of nilpotent elements in a borel subalgebra of $g$. Let $I\_{g}$ be the ideal of polynomial functions on $g\times g$ generated by the functions whose value in $(x,y)$ is the scalar product of $v$ and $[x,y]$ where $v$ is in $g$. The main result is the theorem: Let us suppose that for any semi-simple element in $g$, the simple factors of its centralizer in $g$ have the property (N). Then the complex $E\_{\bullet}(g)$ has no homology in degree different from b and its homology in degree b is the reduced algebra of regular functions on the commuting variety. In particular, $I\_{g}$ is a prime ideal whose set of zeros in $g\times g$ is the commuting variety of $g$.
The main theorem (Theorem 4.1) of this paper claims that any ring morphism from an Azumaya algebra of constant rank over a commutative ring to another one of the same constant rank and over a reduced commutative ring induces a ring morphism between the centers of these algebras. The second main theorem (Theorem 5.3) implies (through its Corollary 5.4) that a ring morphism between two Azumaya algebras of the same constant rank is an isomorphism if and only if it induces an isomorphism between their centers. As preliminary to these theorems we also prove a theorem (Theorem 2.8) describing the Brauer group of a commutative Artin ring and give an explicit proof of the converse of the Artin-Procesi theorem on Azumaya algebras (Theorem 2.6).