arXiv subjects
Publications and source records attributed to François Dumas.
We study algebraic properties of the super Jordan plane~$\mathcal B$ introduced by I. Angiono, I. Heckenberger and the first named author. Concretely we show that $\mathcal B$ is super-prime and has a super-simple super-artinian ring of fractions. We also compute the groups of superalgebra and braided Hopf algebra automorphisms. Instrumental for our approach is the embedding of the Jordan plane as a subalgebra of $\mathcal B$.
The notion of double depth associated with quasi-Jacobi forms allows distinguishing, within the algebra of quasi-Jacobi singular forms of index zero, certain significant subalgebras (modular-type forms, elliptic-type forms, Jacobi forms). We study the stability of these subalgebras under the derivations of this algebra and through certain sequences of bidifferential operators constituting analogs of Rankin-Cohen brackets or transvectants
The notion of double depth associated with quasi-Jacobi forms allows distinguishing,within the algebra of quasi-Jacobi singular forms of index zero, certain significant subalgebras (modular-type forms, elliptic-type forms, Jacobi forms). We study the stability of these subalgebras under the derivations of the algebra of quasi-Jacobi singular forms of index zero and through certain sequences of bidifferential operators constituting analogs of Rankin-Cohen brackets or transvectants.
We study from the point of view of rational equivalence the enveloping algebras of Lie algebras of dimension 3 whose derived Lie subalgebra is of dimension 2, over an algebraically closed base field in arbitrary characteristics.
We compute the simple finite-dimensional modules and the center of the Drinfeld double of the Jordan plane introduced in $\texttt{arXiv:2002.02514}$ assuming that the characteristic is zero.
This work is devoted to the algebraic and arithmetic properties of Rankin-Cohen brackets allowing to define and study them in several natural situations of number theory. It focuses on the property of these brackets to be formal deformations of the algebras on which they are defined, with related questions on restriction-extension methods. The general algebraic results developed here are applied to the study of formal deformations of the algebra of weak Jacobi forms and their relation with the Rankin-Cohen brackets on modular and quasimodular forms.
We study from an algebraic point of view the question of extending an action of a group \(Γ\) on a commutative domain \(R\) to a formal pseudodifferential operator ring \(B=R(\!(x\,;\,d)\!)\) with coefficients in \(R\), as well as to some canonical quadratic extension \(C=R(\!(x^{1/2}\,;\,\frac 12 d)\!)_2\) of \(B\). We give a necessary and sufficient condition of compatibility between the action and the derivation $d$ of $R$ for such an extension to exist, and we determine all possible extensions of the action to \(B\) and \(C\). We describe under suitable assumptions the invariant subalgebras \(B^Γ\) and \(C^Γ\) as Laurent series rings with coefficients in \(R^Γ\). The main results of this general study are applied in a numbertheoretical context to the case where \(Γ\) is a subgroup of \({\rm SL}(2,\C)\) acting by homographies on an algebra \(R\) of functions in one complex variable. Denoting by \(M_j\) the vector space of algebraic modular forms in $R$ of weight \(j\) (even or odd), we build for any nonnegative integer \(k\) a linear isomorphism between the subspace \(C_k^Γ\) of invariant operators of order \(\geq k\) in \(C^Γ\) and the product space \(\mathcal{M}_k=\prod_{j\geq k}M_j\), which can be identified with a space of algebraic Jacobi forms of weight \(k\). It results in particular a structure of noncommutative algebra on \(\mathcal M_0\) and an algebra isomorphism \(Ψ:\mathcal M_0\to C_0^Γ\), whose restriction to the particular case of even weights was previously known in the litterature. We study properties of this correspondence combining arithmetical arguments and the use of the algebraic results of the first part of the article.
We determine the skew fields of fractions of the enveloping algebra of the Lie superalgebra osp(1, 2) and of some significant subsu-peralgebras of the Lie superalgebra osp(1, 4). We compare the kinds of skew fields arising from this "super" context with the Weyl skew fields in the classical Gelfand-Kirillov property.
We construct and classify all Poisson structures on quasimodular forms that extend the one coming from the first Rankin-Cohen bracket on the modular forms. We use them to build formal deformations on the algebra of quasimodular forms.
Let g be a simple complex finite dimensional Lie algebra and let U_q^+(g) be the positive part of the quantum enveloping algebra of g. If g is of type A_2, the group of algebra automorphisms of U_q^+(g) is a semidirect product of a 2-dimensional torus and the group of automorphisms of the Dynkin diagram; any algebra automorphism is an automorphism of braided Hopf algebra, and preserves the standard grading [AD1]. This intriguing smallness of the group of algebra automorphisms raises questions about the extent of these phenomena. We discuss some of them in the present paper. We introduce the notion of ``algebra with few automorphisms" and establish some consequences. We prove some exploratory results concerning the group of algebra automorphisms for the type B_2. We study Hopf algebra automorphisms of Nichols algebras and their bosonizations and compute in particular the group of Hopf algebra automorphisms of U_q^+(g).