Anti-associative algebras

TL;DR

This paper explores anti-associative algebras, a class of nilpotent nonassociative algebras, describing their free structures, related algebra classes, and associated operads and cohomology complexes.

Contribution

It provides a detailed description of free anti-associative algebras, links them to related algebra classes, and analyzes their operads and deformation cohomology.

Findings

Characterization of free anti-associative algebras

Connections to Jacobi-Jordan algebras and deformation quantization

Description of operads and cohomology complexes for these algebras

Abstract

An anti-associative algebra is a nonassociative algebra whose multiplication satisfies the identity a(bc)+(ab)c=0. Such algebras are nilpotent. We describe the free anti-associative algebras with a finite number of generators. Other types of nonassociative algebras, obtained either by the process of polarization, such as Jacobi-Jordan algebras, or obtained by deformation quantization, are associated with this class of algebras. Following Markl-Remms work, we describe the operads associated with these algebra classes and in particular the cohomology complexes in relation to deformations.