Quandles from gauge transformations

TL;DR

This paper explores the construction of quandle structures from gauge transformations, including principal bundles and their generalizations, linking them to Alexander quandles and Lie/Noether quandles.

Contribution

It introduces new quandle structures derived from gauge transformations and principal bundles, connecting them to existing algebraic frameworks.

Findings

Constructed a quandle from a principal bundle and its discrete version.

Showed the quandle reduces to a generalized Alexander quandle in a special case.

Developed Lie and Noether quandle structures from smooth gauge transformations.

Abstract

In this paper, we investigate a quandle structure induced by an augmented rack arising from a gauge transformation group. We construct a quandle from a principal bundle and its discrete generalization. When we see a group as a (discrete) principal bundle over a point, this quandle becomes equivalent to the generalized Alexander quandle for its inner automorphism. Moreover, we construct a Lie and Noether quandle structure from a smooth gauge transformation.