Braces and Poisson additivity

TL;DR

This paper establishes a connection between brace constructions and additivity functors, showing how certain algebraic structures relate and providing new insights into derived coisotropic structures.

Contribution

It constructs a functor linking brace algebras to associative algebras in homotopy $O$-algebras and identifies categories of $P_{n+1}$-algebras with associative $P_n$-algebras.

Findings

Identifies the category of $P_{n+1}$-algebras with associative $P_n$-algebras.

Shows equivalence of two definitions of derived coisotropic structures.

Relates brace constructions to additivity functors in operad theory.

Abstract

We relate the brace construction introduced by Calaque and Willwacher to an additivity functor. That is, we construct a functor from brace algebras associated to an operad $O$ to associative algebras in the category of homotopy $O$-algebras. As an example, we identify the category of $P_{n+1}$-algebras with the category of associative algebras in $P_n$-algebras. We also show that under this identification there is an equivalence of two definitions of derived coisotropic structures in the literature.