n-ary Lie and Associative Algebras

TL;DR

This paper introduces n-ary Lie and associative algebras using multigraded brackets, providing foundational formulas for their cohomologies, expanding algebraic structures beyond binary operations.

Contribution

It defines n-ary associative and Lie algebras and modules using multigraded brackets, offering new algebraic frameworks and cohomology formulas.

Findings

Defines n-ary associative and Lie algebras

Provides formulas for Hochschild and Chevalley cohomology

Extends algebraic structures beyond binary operations

Abstract

With the help of the multigraded Nijenhuis-- Richardson bracket and the multigraded Gerstenhaber bracket from [7] for every $n\ge 2$ we define $n$-ary associative algebras and their modules and also $n$-ary Lie algebras and their modules, and we give the relevant formulas for Hochschild and Chevalley cohomogy.