Some non-Koszul algebras from rational homotopy theory

Abstract

The McCool group, denoted $P{\Sigma }_n$, is the group of pure symmetric automorphisms of a free group of rank $n$. The cohomology algebra $H^{\ast }(P{\Sigma }_n,\mathbb{Q})$ was determined by Jensen, McCammond and Meier. We prove that $H^{\ast }(P{\Sigma }_n,\mathbb{Q})$ is a non-Koszul algebra for $n\ge 4$, 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.

10 pages; revised version to appear in The Bulletin of the London Mathematical Society

References in corpus (2)

• Algebraic invariants for right-angled Artin groups
• The integral cohomology of the group of loops