Cumulants, Koszul brackets and homological perturbation theory for commutative $BV_\infty$ and $IBL_\infty$ algebras

TL;DR

This paper investigates the connections between cumulants and Koszul brackets, demonstrating their compatibility with homological perturbation theory, and introduces new transfer theorems and morphisms for advanced algebraic structures.

Contribution

It establishes the exponential relationship between cumulants and Koszul brackets, proves new homotopy transfer theorems for commutative BV_infinity algebras, and introduces a novel definition of morphisms for these algebras.

Findings

Cumulants are an exponential version of Koszul brackets.

Compatibility of cumulants and brackets with homological perturbation theory.

New homotopy transfer theorem for commutative BV_infinity algebras.

Abstract

We explore the relationship between the classical constructions of cumulants and Koszul brackets, showing that the former are an expontial version of the latter. Moreover, under some additional technical assumptions, we prove that both constructions are compatible with standard homological perturbation theory in an appropriate sense. As an application of these results, we provide new proofs for the homotopy transfer Theorem for $L_\infty$ and $IBL_\infty$ algebras based on the symmetrized tensor trick and the standard perturbation Lemma, as in the usual approach for $A_\infty$ algebras. Moreover, we prove a homotopy transfer Theorem for commutative $BV_\infty$ algebras in the sense of Kravchenko which appears to be new. Along the way, we introduce a new definition of morphism between commutative $BV_\infty$ algebras.