We determine the multiplicative structures of the homotopy operations on derived Lie algebras, derived restricted Lie algebras and partition Lie algebras, as well as the relation between these. Under Koszul duality, this follows from a calculation of the homotopy operations on derived rings, p-truncated algebras and divided power algebras over Fₚ.
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Nuiten et al. (2026) studied this question.
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