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Some non-Koszul algebras from rational homotopy theory

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Andrew Conner, Peter Goetz

Rings and Algebras · Submitted 17 Jul 2014

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim 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.

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arXiv ID1407.4726 (version 2)
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CitationAndrew Conner, Peter Goetz. "Some non-Koszul algebras from rational homotopy theory." arXiv:1407.4726 (2014). https://pith.science/paper/I2IROBOL
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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,710 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.12 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.79 · compares with prior results 0.56 · mentions code or data availability 0.01.

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arXiv listing abstract v2. Results and the main claim are underlined; .

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.

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#sentencerolethe authors' own workstates the main conclusionstrength
s0 The McCool group, denoted $P\Sigma_n$, is the group of pure symmetric automorphisms of a free group of rank $n$. background 1.00 0.05 0.01 asserted 1.00
s1 The cohomology algebra $H^*(P\Sigma_n, \mathbb{Q})$ was determined by Jensen, McCammond and Meier. background 1.00 0.03 0.02 asserted 1.00
s2 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. result 0.98 0.97 0.91 asserted 1.00
s3 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. contribution 0.57 0.96 0.05 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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Original 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 $H^*(P\Sigma_n, \mathbb{Q})$ 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.

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