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Article

Konovalov, Nikolay
Koszul duality for simplicial restricted Lie algebras. (English). Higher Structures, vol. 8 (2024), issue 2, pp. 248-331
MSC: 16T15, 17B56, 18M70, 18N50, 55S10 | MR 4835391 | Zbl 08006153 | DOI: 10.21136/HS.2024.12
Keywords:
Koszul duality; restricted Lie algebras; truncated coalgebras; $\Lambda$-algebra
Summary:
Let $s_0$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 Ho$(s_0Lie_(ξ)^(r))$ of the homotopy category Ho$s_0Lie^(r)$ and an equivalence Ho$(s_0Lie_(ξ)^(r))$ $\simeq$ Ho$(s_1CoAlg^(tr))$. Here $s_1CoAlg^(tr)$ 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 $\pi_{\infty}(L_(•))$ for $L_(•) \in Ho(s_0Lie^(r))$ to Lie in the full subcategory Ho$(s_0Lie_(ξ)^(r))$. 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 sLie$^r$. We use this spectral sequence to recompute the homotopy groups of a free simplicial restricted Lie algebra.
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