On classification of irreducible quandle modules over a connected quandle

Abstract

We study modules over quandles and classify irreducible quandle modules. The main result of this paper states that there is a correspondence between irreducible modules over a quandle $Q$ and irreducible modules over certain groups: more specifically, irreducible modules over the fundamental group of $Q$ and nontrivial irreducible modules over the associated group $\text{As}(Q)$. As an application, we classify irreducible modules over generalized dihedral quandles, the quandles obtained from generalized dihedral groups, and connected quandles in $S{L}_2({\mathbb{F}}_q)$ where ${\mathbb{F}}_q$ denotes the finite field of $q=p^f$ elements.