  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.
