Universal Lie Formulas for Higher Antibrackets

TL;DR

This paper introduces universal formulas for higher antibrackets of linear operators on superalgebras, enabling their extension to noncommutative algebras while maintaining key algebraic identities.

Contribution

It provides a universal construction for higher antibrackets using Nijenhuis-Richardson brackets, extending their applicability to noncommutative algebras.

Findings

Universal formulas for higher antibrackets derived

Extension to noncommutative algebras achieved

Preservation of generalized Jacobi identities

Abstract

We prove that the hierarchy of higher antibrackets (aka higher Koszul brackets, aka Koszul braces) of a linear operator $\Delta$ on a commutative superalgebra can be defined by some universal formulas involving iterated Nijenhuis-Richardson brackets having as arguments $\Delta$ and the multiplication operators. As a byproduct, we can immediately extend higher antibrackets to noncommutative algebras in a way preserving the validity of generalized Jacobi identities.