Derivations of quandles

TL;DR

This paper develops a theory of derivations for quandles, introducing a universal derived quandle and exploring its properties, including structure and extensions to virtual quandles.

Contribution

It introduces the concept of derivations for quandles, constructs a universal derived quandle, and extends the theory to virtual quandles, advancing quandle theory foundations.

Findings

Existence of a unique derived quandle for each quandle.

Derivations form an abelian quandle when target is abelian.

Extension of derivation concepts to virtual quandles.

Abstract

The aim of this paper is to propose a theory of derivations for quandles. Given a quandle $A$ admitting an action by a quandle $Q$, derivations from $Q$ to $A$ are introduced as twisted analogues of quandle homomorphisms. It is shown that for each quandle $Q$ there exists a unique $Q$-quandle $\mathcal{A}_Q$ (the derived quandle of $Q$) such that derivations from $Q$ to any $Q$-quandle $A$ are in bijective correspondence with $Q$-quandle homomorphisms from $\mathcal{A}_Q$ to $A$. Further, it is proved that the set of all derivations to an abelian $Q$-quandle $A$ has the structure of an abelian quandle, and inherits many other properties from $A$. In the end, the ideas are extended to the setting of virtual quandles.