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Norbert Poncin

Publications and source records attributed to Norbert Poncin.

42 records · Page 3Linked to original sources
arXiv2006

Formal Poisson cohomology of twisted $r$-matrix induced structures

Quadratic Poisson tensors of the Dufour-Haraki classification read as a sum of an $r$-matrix induced structure twisted by a (small) compatible exact quadratic tensor. An appropriate bigrading of the space of formal Poisson cochains then leads to a vertically positive double complex. The associated spectral sequence allows to compute the Poisson-Lichnerowicz cohomology of the considered tensors. We depict this modus operandi, apply our technique to concrete examples of twisted Poisson structures, and obtain a complete description of their cohomology. As richness of Poisson cohomology entails computation through the whole spectral sequence, we detail an entire model of this sequence. Finally, the paper provides practical insight into the operating mode of spectral sequences.

math.SG↗
arXiv2005

On quantum and classical Poisson algebras

Results on derivations and automorphisms of some quantum and classical Poisson algebras, as well as characterizations of manifolds by the Lie structure of such algebras, are revisited and extended. We prove in particular somehow unexpected fact that the algebras of linear differential operators acting on smooth sections of two real vector bundles of rank 1 are isomorphic as Lie algebras if and only if the base manifolds are diffeomorphic, independently whether the line bundles themselves are isomorphic or not.

math.DG↗
arXiv2005

Lie algebraic characterization of manifolds

Results on characterization of manifolds in terms of certain Lie algebras growing on them, especially Lie algebras of differential operators, are reviewed and extended. In particular, we prove that a smooth (real-analytic, Stein) manifold is characterized by the corresponding Lie algebra of linear differential operators, i.e. isomorphisms of such Lie algebras are induced by the appropriate class of diffeomorphisms of the underlaying manifolds.

math.DG↗
arXiv2002

Automorphisms of quantum and classical Poisson algebras

We prove Pursell-Shanks type results for the Lie algebra D(M) of all linear differential operators of a smooth manifold M, for its Lie subalgebra D^1(M) of all linear first-order differential operators of M, and for the Poisson algebra S(M)=Pol(T*M) of all polynomial functions on the cotangent bundle T*M, the symbols of the operators in D(M). Chiefly however we provide explicit formulas describing completely the automorphisms of the Lie algebras D^1(M), S(M), and D(M).

math.RA↗
arXiv2002

Equivariant Operators between some Modules of the Lie Algebra of Vector Fields

The space D(k,p) of differential operators of order at most k, from the differential forms of degree p of a smooth manifold M into the functions of M, is a module over the Lie algebra of vector fields of M, when it's equipped with the natural Lie derivative. In this paper, we compute all equivariant i.e. intertwining operators from D(k,p) into D(k',p') and conclude that the preceding modules of differential operators are never isomorphic. We also answer a question of P. Lecomte, who observed that the restriction to D(k,p) of some homotopy operator, introduced in one of his works, is equivariant for small values of k and p.

math.RT↗