The derivation problem for quandle algebras

TL;DR

This paper introduces and characterizes derivations in quandle algebras, providing explicit classifications for dihedral and Alexander quandles, and exploring their inner derivations and symmetry properties.

Contribution

It offers a comprehensive analysis of derivations in quandle algebras, including explicit classifications for dihedral and Alexander quandles, and investigates their inner derivations.

Findings

Complete characterization of derivations for dihedral quandle algebras over characteristic zero fields.

Explicit examples and computations of derivations over various characteristics.

Necessary conditions and low-dimensional computations for inner derivations in Alexander quandle algebras.

Abstract

The purpose of this paper is to introduce and investigate the notion of derivation for quandle algebras. More precisely, we describe the symmetries on structure constants providing a characterization for a linear map to be a derivation. We obtain a complete characterization of derivations in the case of quandle algebras of \emph{dihedral quandles} over fields of characteristic zero, and provide the dimensionality of the Lie algebra of derivations. Many explicit examples and computations are given over both zero and positive characteristic. Furthermore, we investigate inner derivations, in the sense of Schafer for non-associative structures. We obtain necessary conditions for the Lie transformation algebra of quandle algebras of Alexander quandles, with explicit computations in low dimensions.