Abstract:

The usual PBW theorem states that for a Lie algebra $g$ over a field of characteristic zero the associated graded algebra of the universal enveloping $Ug$ with respect to the degree filtration is isomorphic to the symmetric algebra of $g$. The completed PBW theorem concerns the structure of the completion of $Ug$ in the augmentation ideal. The simplest examples shows that the completed PBW theorem does not hold for general Lie algebras. We'll show that it, nevertheless, holds for nilpotent Lie algebras. As a consequence, we'll prove that the cohomology algebra $H(g)$ of a nilpotent Lie algebra is generated by $H^1(g)$ by (suitably defined) Massey operations. We'll also discuss why the more natural setting for this problem is that of coalgebras instead of algebras and some consequences of computations of cohomology of nilpotent Lie algebras.