Burnside rings for racks and quandles

TL;DR

This paper introduces Burnside rings for finite racks and quandles, providing a new algebraic framework that enhances classification and connects with broader algebraic theories.

Contribution

It develops the structure of Burnside rings for racks and quandles, including bases, generators, and a theory of marks, advancing their classification methods.

Findings

Established additive bases and multiplicative generators for the Burnside rings.

Developed a comprehensive theory of marks to distinguish elements.

Extended the Dress--Siebeneicher theory to infinite cyclic groups.

Abstract

We restructure and advance the classification theory of finite racks and quandles by employing powerful methods from transformation groups and representation theory, especially Burnside rings. These rings serve as universal receptacles for those invariants of racks and quandles that are additive with respect to decompositions. We present several fundamental results regarding their structure, including additive bases and multiplicative generators. We also develop a theory of marks, which is analogous to counting fixed points of group actions and computing traces in character theory, and which is comprehensive enough to distinguish different elements in the Burnside rings. The new structures not only offer a fresh framework for the classification theory of finite racks and quandles but also equip us with tools to develop these ideas and create interfaces that strengthen connections with related areas of algebra. For example, they extend the Dress--Siebeneicher theory of the Burnside ring of the infinite cyclic group beyond the realm of permutation racks.