François Dumas

Invariants du corps de weyl sous l'action de groupes finis

Let R be an iterated Ore extension k[y][x;σ,δ] of the complex number field k, with δ a k-automorphism of k[y] and δ a u-derivation of k[y] vanishing on k. We suppose that the center of R is k. Up to a change of variables, any finite group G of k-automorphisms of R acts linearly on kx⊕D1ky. When the quotient division ring D of R is isomorphic to the Weyl skewfield D1(k)1 , then DG⋍D1 (k). In any other noncommutative case, D is isomorphic to the quantum Weyl skewfield Dq 1(k) for some q∊k∗ not a root of one, and DG⋍Ds 1(k) with s = q‖G‖.

PRIME SPECTRUM AND AUTOMORPHISMS FOR 2×2 JORDANIAN MATRICES

ABSTRACT This paper is devoted to some ring theoretic properties of the jordanian deformation of the algebra of regular functions on the matrices with coefficients in an algebraically closed field of characteristic zero, and of the associated factor algebra . We prove in particular that the prime spectrum of is the disjoint union of five components, each of which being homeomorphic to the spectrum of a commutative (possibly localised) polynomial ring. So we can give an explicit description of the prime spectrum of , and check that any prime factor of satisfies the Gelfand-Kirillov property. Then we study the automorphism groups of the algebras and and prove that they are generated by linear automorphisms and exponentials of locally nilpotent derivations.

Poisson structures and star products on quasimodular forms

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.

Opérateurs différentiels invariants et problème de Noether

Let G be a group and ≤ : G → GL(V) a representation of G in a vector space V of dimension n over a commutative field k of characteristic zero. The group ≤(G) acts by automorphisms on the algebra of regular functions k[V], and this action can be canonically extended to theWeyl algebra A n (k) of differential operators over k[V] and then to the skewfield of fractions D n(k) of A n(k). The problem studied in this paper is to determine sufficient conditions for the subfield of invariants of D n(k) under this action to be isomorphic to a Weyl skewfield D m(K) for some integer 0 ≤ m ≤ n and some purely transcendental extension K of k. We obtain such an isomorphism in two cases: (1) when ≤ splits into a sum of representations of dimension one, (2) when ≤ is of dimension two. We give some applications of these general results to the actions of tori on Weyl algebras and to differential operators over Kleinian surfaces.

Field Generators for the Quantum Plane

Let k be a commutative field and q a (nonzero and not root of one) quantization parameter in k. Manin’s quantum plane P = k q[x,y] is the k-algebra of noncommutative polynomials in two variables with commutation law xy = qyx. The quantum torus R = k q[x ±1, y ±1 ] is the simple localization of P consisting of quantum Laurent polynomials. We denote by k q(x,y) = Frac R = Frac P the skew field of quantum rational functions over k. For any nonzero polynomials A,B ∈R such that AB = q BA, the (skew) subfield k q(A, B) of k q(x, y) generated by A and B is isomorphic to k q(x,y); the main question discussed in the paper is then: do we have k q(x,y) = k q(A,B)? We prove that this equality holds if at least one of the generators A or B is a monomial in R, or if the support of at least one of them is based on a line.

Skew power series rings with general commutation formula

Abstract The noncommutative product in a skew power series ring Λ in an indeterminate X with coefficients in a skewfield K is entirely determined from a commutation law between X and any element of K . Different types of commutation rules are described, with some resulting properties for Λ and its skewfield of fractions.