Constructing an infinite family of quandles from a quandle

Abstract

Quandles are self-distributive, right-invertible, idempotent algebras. A group with conjugation for binary operation is an example of a quandle. Given a quandle $(Q, \ast)$ and a positive integer $n$, define $a\ast_n b = (\cdots (a\ast \underbrace{b)\ast \cdots )\ast b}_{n}$, where $a, b \in Q$. Then, $(Q, \ast_n)$ is again a quandle. We set forth the following problem. ``Find $(Q, \ast)$ such that the sequence $\{(Q, \ast_n)\, :\, n\in \mathbb{Z}^+ \}$ is made up of pairwise non-isomorphic quandles.'' In this article we find such a quandle $(Q, \ast)$. We study the general linear group of $2$-by-$2$ matrices over $\mathbb{C}$ as a quandle under conjugation. Its (algebraically) connected components, that is, its conjugacy classes, are subquandles of it. We show the latter are connected as quandles and prove rigidity results about them such as the dihedral quandle of order $3$ is not a subquandle for most of them. Then we consider the quandle which is the projective linear group of $2$-by-$2$ matrices over $\mathbb{C}$ with conjugation, and prove it solves the problem above. In the course of this work we prove a sufficient and necessary condition for a quandle to be latin. This will reduce significantly the complexity of algorithms for ascertaining if a quandle is latin.