On classification of irreducible quandle modules over a connected quandle (Q6039907)

Quandles have emerged to be an important tool in constructing knot invariants in recent years. A quandle is an algebraic object whose binary operation \(\rhd\) is motivated by the famous Reidemeister moves (from knot theory). Any group \(G\) gives rise to the conjugation quandle, defined as \(g\rhd h= ghg^{-1}\). This defines a functor \(\mathrm{Conj}: \mathbf{Grp} \longrightarrow \mathbf{Qd}\), where \(\mathbf{Grp}\), \(\mathbf{Qd}\) denote the categories of groups and quandles respectively. For a quandle \(Q\), the group defined by the presentation \[\mathrm{As}(Q)=\langle g_q~(q\in Q)|g_{p\rhd q}=g_pg_qg_p^{-1}\rangle\] is called the \textit{associated group of \(Q\)}. This defines another functor \(\mathrm{As}:\mathbf{Qd} \longrightarrow \mathbf{Grp}\) in the other direction. It is known from Proposition 2.1 of [\textit{R. Fenn} and \textit{C. Rourke}, J. Knot Theory Ramifications 1, No. 4, 343--406 (1992; Zbl 0787.57003)] that the functor \(\mathrm{As}\) is a left adjoint of \(\mathrm{Conj}\). Thus a module over \(\mathrm{As}(Q)\) becomes a \(Q\) module. Although not all quandle modules are of this form, hence the classification of quandle modules requires more effort. The paper under review considers the problem of classifying irreducible modules over \textit{connected} quandles. The strategy is as follows. For a quandle \(Q\) and a \(Q\)-module \(\mathscr{M}\), first, consider the inner-automorphism group \(\mathrm{Inn}(\mathscr{M})\) as the conjugation quandle. Then one can construct another quandle module \(\mathscr{I}(\mathscr{M})\) from \(\mathrm{Inn}(\mathscr{M})\), over \(Q\), which is induced from an \(\mathrm{As}(Q)\)-module and a homomorphism \(i_{\mathscr M}:\mathscr M\longrightarrow \mathscr I(\mathscr M)\). If \(\mathscr M\) is irreducible then either \(i_{\mathscr M}\) is injective or \(0\). Denote the commutator subgroup of \(\mathrm{As}(Q)\) by \(\mathrm{As}_0(Q)\) and take \(\mathrm{As}_q(Q)=\{x\in\mathrm{As}{Q}|x\cdot q=q\}\) for \(q\in Q\). Define the \textit{fundamental group of \(Q\) at \(q\)} to be \(\pi_1(Q,q)=\mathrm{As}_q(Q)\cap \mathrm{As}_0(Q)\). Then the two main results are as follows. \begin{itemize} \item An irreducible module \(\mathscr M\) such that \(i_{\mathscr M}\) is zero corresponds to an irreducible module over a group \(\pi_1(Q,q)\)''. \item Otherwise, \(\mathscr M\) corresponds to an irreducible \(\mathrm{As}(Q)\) module in some way. \end{itemize} As an application of the main results, the author further gives explicit descriptions of \begin{itemize} \item irreducible modules over generalized dihedral quandles with coefficients in fields of characteristic zero, \item irreducible modules over connected quandle \(Q\) in the special linear group \(\mathrm{SL}_2(\mathbb{F}_q)\) with coefficients in certain fields of characteristic \(p\), where \(q=p^f\) for some \(f\geq 1\). \end{itemize}