Jacques Alev

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‖.

A class of counterexamples to the Gel’fand-Kirillov conjecture

LIMBURGS UNIV CENTRUM,DEPT WNI,B-3500 DIEPENBEEK,BELGIUM.Alev, J, UNIV REIMS,UFR SCI,DEPT MATH,MOULIN HOUSSE,BP 347,F-51062 REIMS,FRANCE.

Comparaison de L’Homologie de Hochschild et de L’Homologie de Poisson Pour Une Deformation des Surfaces de Klein

Let P G be the quotient variety of the affine plane by the action of a finite group G ⊂ SL(2,ℂ); then P G inherits in a natural way a Poisson algebra structure. Let A 1 (ℂ) be the first Weyl algebra ℂ[p, q] with the relation pq-qp=1, on which G acts by automorphisms in such a way that the invariant algebra A 1 (ℂ) G is a deformation of P G . We prove that the trace group HH 0(A 1(ℂ) G ) is a deformation of the Poisson homology group HH 0(A 1(ℂ) G ). Moreover, these two groups are ℂ-vector spaces of finite dimension and dim (HH 0(A 1(ℂ) G )) = dim (H 0 Pois (P G )) = s(G) - 1, where s(G) denotes the number of irreducible representations of G.

Finite Group Actions on Poisson Algebras

Let Andenote the Weyl algebra of all differential operators on the polynomial algebra C[X1,… Xn].It is well known that if G is a finite group of algebra automorphisms of An, then An is a simple algebra. (See [12] pp. 20–23 for an algebraic proof or [15] Lemma 1.2 for an analytic approach.) It is natural to expect that the analogous result holds for the associated graded object. To be precise, if Anis filtered by total degree, then the associated graded algebra is the larger polynomial ring R = C[X1, …Xn,Y1,… Yn]with the Poisson bracket which describes a standard symplectic affine space. To be explicit R is also a Lie algebra subject to

Le Groupe des Traces de Poisson de Certaines Algèbres D'invariants

Let 𝔤 be a simple Lie algebra of rank l, 𝔥 a Cartan subalgebra and W the corresponding Weyl group. The space 𝔥⊕𝔥* is naturally equipped with a symplectic structure and the invariant algebra S(𝔥⊕𝔥*) W inherits a Poisson structure which can be deformed by the invariant algebra where A l denotes the Weyl algebra of rank l. The purpose of this article is to compute the Poisson homology group of degree zero of S(𝔥⊕𝔥*) W and to compare it to the Hochschild homology group of degree zero of . These two groups have the same dimensions in rank 2 which are equal respectively to 1.2 and 3 in the types A 2, B 2 and G 2.

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.