# Operadic approach to cohomology of associative triple and N-tuple systems

## TL;DR

This paper explores the structure of cohomology for n-ary totally associative algebras using operadic methods, defining a cup product that gives the cohomology an n-ary partially associative algebra structure.

## Contribution

It introduces an operadic approach to define and analyze the cup product in the cohomology of n-ary associative algebras, especially for n=3 and 4.

## Key findings

- Defined explicit cup product for n=3 and 4
- Proved basis properties of the cup product
- Established n-ary partial associativity in cohomology

## Abstract

The cup product in the cohomology of algebras over quadratic operads has been studied in the general setting of Koszul duality for operads. We study the cup product on the cohomology of n-ary totally associative algebras with an operation of even (homological) degree. This cup product endows the cohomology with the structure of an n-ary partially associative algebra with an operation of even or odd degree depending on the parity of n. In the cases n = 3 and n = 4, we provide an explicit definition of this cup product and prove its basis properties.