Divided power algebras and distributive laws

TL;DR

This paper develops a systematic framework for understanding divided power structures over products of operads with distributive laws, including specific algebraic cases in various characteristics.

Contribution

It introduces a method to derive divided power algebras over product operads from their factor structures, covering derivations, p-level, and Poisson algebras.

Findings

Characterization of divided power algebras over product operads

Description of divided power algebras with operadic derivation

Analysis of divided power p-level and Poisson algebras in specific characteristics

Abstract

We study the divided power structures over a product of operads with distributive law. We give a systematic method to characterise the divided power algebras over such a product from the structures of divided power algebra coming from each of the the factor operads. We characterise divided power algebras with operadic derivation, as well as divided power $p$-level algebras in characteristic $p$, and divided power Poisson algebras in characteristic 3.