Finitely stable racks and rack representations

TL;DR

This paper introduces finitely stable racks, explores their properties, characterizes certain types, and develops a representation theory analogous to Abelian groups, including duality concepts.

Contribution

It defines finitely stable racks, characterizes Alexander quandles, and develops a new representation theory with duality for finite involutive racks.

Findings

Finitely stable racks share properties with Abelian groups.

Characterization of finitely stable Alexander quandles.

Introduction of rack duality analogous to Pontryagin duality.

Abstract

We define a new class of racks, called finitely stable racks, which, to some extent, share various flavors with Abelian groups. Characterization of finitely stable Alexander quandles is established. Further, we study twisted rack dynamical systems, construct their cross-products, and introduce representation theory of racks and quandles. We prove several results on the {\em strong} representations of finite connected involutive racks analogous to the properties of finite Abelian groups. Finally, we define the {\em Pontryagin} dual of a rack as an Abelian group which, in the finite involutive connected case, coincides with the set of its strong irreducible representations.