  The McCool group, denoted $P\Sigma_n$, is the group of pure symmetric
automorphisms of a free group of rank $n$. The cohomology algebra
$H^*(P\Sigma_n, \mathbb{Q})$ was determined by Jensen, McCammond and Meier. We
prove that $H^*(P\Sigma_n, \mathbb{Q})$ is a non-Koszul algebra for $n \geq 4$,
which answers a question of Cohen and Pruidze. We also study the enveloping
algebra of the graded Lie algebra associated to the lower central series of
$P\Sigma_n$, and prove that it has two natural decompositions as a smash
product of algebras.
