Some non-Koszul algebras from rational homotopy theory

The McCool group, denoted $PΣ_n$, is the group of pure symmetric automorphisms of a free group of rank $n$. The cohomology algebra $H^*(PΣ_n, \mathbb{Q})$ was determined by Jensen, McCammond and Meier. We prove that $H^*(PΣ_n, \mathbb{Q})$ 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 $PΣ_n$, and prove that it has two natural decompositions as a smash product of algebras.

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