Restricted $L_\infty$-algebras and a derived Milnor-Moore theorem

Abstract

For every stable presentably symmetric monoidal $\infty$-category $\mathcal{C}$ we use the Koszul duality between the spectral Lie operad and the cocommutative cooperad to construct an enveloping Hopf algebra functor $\mathcal{U}: \mathrm{Alg}_{\mathrm{Lie}}(\mathcal{C}) \to \mathrm{Hopf}(\mathcal{C})$ from spectral Lie algebras in $\mathcal{C}$ to cocommutative Hopf algebras in $\mathcal{C}$ left adjoint to a functor of derived primitive elements. We prove that if $\mathcal{C}$ is a rational stable presentably symmetric monoidal $\infty$-category, the enveloping Hopf algebra functor is fully faithful. We conclude that Lie algebras in $\mathcal{C}$ are algebras over the monad underlying the adjunction $T \simeq \mathcal{U} \circ \mathrm{Lie}: \mathcal{C} \rightleftarrows \mathrm{Alg}_{\mathrm{Lie}}(\mathcal{C}) \to \mathrm{Hopf}(\mathcal{C}), $ where $\mathrm{Lie}$ is the free Lie algebra and $\mathrm{T}$ is the tensor algebra. For general $\mathcal{C}$ we introduce the notion of restricted $L_\infty$-algebra as an algebra over the latter adjunction. For any field $K$ we construct a forgetful functor from restricted Lie algebras in connective $H(K)$-modules to the $\infty$-category underlying a right induced model structure on simplicial restricted Lie algebras over $K $.