Anti-associative dendriform algebras

TL;DR

This paper introduces anti-associative dendriform algebras, exploring their structure, operadic foundations, and connections with anti-associative algebras and Connes cocycles.

Contribution

It defines anti-associative dendriform algebras, studies their properties via operadic methods, and links them to anti-associative algebras with Connes cocycles.

Findings

Anti-associative dendriform algebras are defined by two operations summing to an anti-associative operation.

$ ext{O}$-operators are used to interpret these algebras.

Compatible structures exist on anti-associative algebras with nondegenerate Connes cocycles.

Abstract

The general operadic approach to splitting algebraic operations was developed in \cite{BBGN}. By splitting the product in a given algebraic variety $\mathcal{C}$, notion of $\mathcal{C}$-dendriform algebras was systematically studied in \cite{OPV}. This article aims to study ``anti-associative dendriform algebras", which offer an approach to addressing anti-associativity. These algebras are defined by two operations whose sum is anti-associative. Furthermore, the notion of $\mathcal{O}$-operators on anti-associative algebras is presented as a tool to interpret anti-associative dendriform algebras. Moreover, anti-associative algebras with nondegenerate Connes cocycles admit compatible anti-associative dendriform algebra structures.