Associator dependent algebras and Koszul duality

TL;DR

This paper classifies all Koszul operads that encode associator dependent algebras, resolving a long-standing open question about their action on octonions and related structures.

Contribution

It provides a comprehensive classification of Koszul operads for associator dependent algebras, including the case of octonions, advancing the understanding of algebraic operad structures.

Findings

Identified all Koszul operads acting on associator dependent algebras.

Solved Loday's open problem regarding octonion algebra actions.

Established a general framework for classifying Koszul operads in this context.

Abstract

We resolve a ten year old open question of Loday of describing Koszul operads that act on the algebra of octonions. In fact, we obtain the answer by solving a more general classification problem: we find all Koszul operads among those encoding associator dependent algebras.