Deformation par quantification et rigidite des algebres enveloppantes

Abstract

We call a finite-dimensional complex Lie algebra $\mathfrak{g}$ strongly rigid if its universal enveloping algebra $\Ug$ is rigid as an associative algebra, i.e. every formal associative deformation is equivalent to the trivial deformation. In quantum group theory this phenomenon is well-known to be the case for all complex semisimple Lie algebras. We show that a strongly rigid Lie algebra has to be rigid as Lie algebra, and that in addition its second scalar cohomology group has to vanish (which excludes nilpotent Lie algebras of dimension greater or equal than two). Moreover, using Kontsevitch's theory of deformation quantization we show that every polynomial deformation of the linear Poisson structure on $\mathfrak{g}^*$ which induces a nonzero cohomology class of $\mathfrak{g}$ leads to a nontrivial deformation of $\mathcal{U}\mathfrak g$. Hence every Poisson structure on a vector space which is zero at some point and whose linear part is a strongly rigid Lie algebra is therefore formally linearizable in the sense of A. Weinstein. Finally we provide examples of rigid Lie algebras which are not strongly rigid, and give a classification of all strongly rigid Lie algebras up to dimension 6.