Racks as multiplicative graphs

TL;DR

This paper introduces a new perspective on racks as multiplicative graphs, developing their homology, algebraic structures, and connections to Lie algebras and Hopf algebras within a categorical framework.

Contribution

It provides a novel interpretation of augmented racks as multiplicative graphs and constructs their algebraic counterparts, linking racks to Hopf and Lie algebras in a categorical setting.

Findings

Racks can be modeled as multiplicative graphs.

A new homology theory for racks is defined.

Discrete racks lead to Hopf and Lie algebras in the Loday-Pirashvili category.

Abstract

We interpret augmented racks as a certain kind of multiplicative graphs and show that this point of view is natural for defining rack homology. We also define the analogue of the group algebra for these objects; in particular, we see how discrete racks give rise to Hopf algebras and Lie algebras in the Loday-Pirashvili category $\mathcal{LM}$. Finally, we discuss the integration of Lie algebras in $\mathcal{LM}$ in the context of multiplicative graphs and augmented racks.