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Spectral Lie Algebras and Bar Towers

Author: Wei Yang · Added: September 20, 2026

Article date: September 2026.

Abstract

We compare two constructions of higher enveloping algebras in a presentably symmetric monoidal stable $\infty$ -category. Chevalley-Eilenberg chains preserve finite products, and the resulting Lie-theoretic cone along actual bar is equivalent to the cone obtained by Koszul dualizing the inclusions of little-disks operads. For $T(h)$ -local spectra, with $T(0)=H\mathbb{Q}$ , we then prove that the induced functor from spectral Lie algebras to compatible bar towers is an equivalence exactly when $h=0$ or $h=1$ . At every height $h\ge2$ and every prime, connective covers of $K(1)$ give a nonzero bar tower annihilated by the right adjoint. The rational case follows from the comparison and also admits an independent proof by free suspensions and shuffle bar.