Racks as multiplicative graphs

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.

Keywords

rack, multiplicative graph, Loday–Pirashvili category, Leibniz algebra

2010 Mathematics Subject Classification

05C99, 16T05, 20N99

Full Text (PDF format)

Received 7 October 2017

Received revised 14 February 2018

Published 13 June 2018