Koszul duality for simplicial restricted Lie algebras

TL;DR

This paper establishes a duality between certain simplicial restricted Lie algebras and truncated coalgebras, providing new tools for computing homotopy groups via an analog of the unstable Adams spectral sequence.

Contribution

It proves an equivalence of homotopy categories linking simplicial restricted Lie algebras and truncated coalgebras, and constructs a spectral sequence for homotopy computations.

Findings

Established a category equivalence between simplicial restricted Lie algebras and truncated coalgebras.

Provided a criterion for homotopy groups to lie in a specific subcategory.

Recomputed homotopy groups of free simplicial restricted Lie algebras using the spectral sequence.

Abstract

Let $\mathsf{s}_0\mathsf{Lie}^r$ be the category of $0$-reduced simplicial restricted Lie algebras over a fixed perfect field of positive characteristic $p$. We prove that there is a full subcategory $\mathrm{Ho}(\mathsf{s}_0\mathsf{Lie}^r_{\xi})$ of the homotopy category $\mathrm{Ho}(\mathsf{s}_0\mathsf{Lie}^r)$ and an equivalence $\mathrm{Ho}(\mathsf{s}_0\mathsf{Lie}^r_{\xi})\simeq\mathrm{Ho}(\mathsf{s}_1\mathsf{CoAlg}^{tr})$. Here $\mathsf{s}_1\mathsf{CoAlg}^{tr}$ is the category of $1$-reduced simplicial truncated coalgebras; informally, a coaugmented cocommutative coalgebra $C$ is truncated if $x^p=0$ for any $x$ from the augmentation ideal of the dual algebra $C^*$. Moreover, we provide a sufficient and necessary condition in terms of the homotopy groups $\pi_*(L_\bullet)$ for $L_\bullet \in \mathrm{Ho}(\mathsf{s}_0\mathsf{Lie}^r)$ to lie in the full subcategory $\mathrm{Ho}(\mathsf{s}_0\mathsf{Lie}^r_{\xi})$. As an application of the equivalence above, we construct and examine an analog of the unstable Adams spectral sequence of A. K. Bousfield and D. Kan in the category $\mathsf{s}\mathsf{Lie}^r$. We use this spectral sequence to recompute the homotopy groups of a free simplicial restricted Lie algebra.