The BV formalism for L$_\infty$-algebras

TL;DR

This paper explores the functorial relationship between BV$_ $-algebras and L$_ $-algebras, establishing categorical equivalences and adjoint functors, and advances the theory of morphisms in these algebraic structures.

Contribution

It characterizes the category of L$_\infty$-algebras via BV$_\infty$-algebras and introduces a left adjoint functor, expanding the understanding of their functorial properties.

Findings

Categorical characterization of L$_\infty$-algebras as pure BV$_\infty$-algebras.

Existence of a left adjoint functor to the higher derived brackets.

Development of the morphism theory with the logarithm of a map.

Abstract

Functorial properties of the correspondence between commutative BV$_\infty$-algebras and L$_\infty$-algebras are investigated. The category of L$_\infty$-algebras with L$_\infty$-morphisms is characterized as a certain category of pure BV$_\infty$-algebras with pure BV$_\infty$-morphisms. The functor assigning to a commutative BV$_\infty$-algebra the L$_\infty$-algebra given by higher derived brackets is also shown to have a left adjoint. Cieliebak-Fukaya-Latschev's machinery of IBL$_\infty$- and BV$_\infty$-morphisms is further developed with introducing the logarithm of a map.