Isomorphism Invariants for Linear Quasigroups

TL;DR

This paper explores isomorphism invariants for $Z$-linear quasigroups, examining how ordinary characters classify them and introducing permutational similarity as a new equivalence concept.

Contribution

It investigates the classification problem for $Z$-linear quasigroups, demonstrating limitations of ordinary characters and proposing permutational similarity as a new invariant.

Findings

Non-isomorphic $Z$-linear quasigroups can share the same ordinary character.

For certain subclasses, equivalences are realized by permutational intertwinings.

Permutational similarity is an intermediate equivalence between isomorphism and isotopy.

Abstract

For a unital ring $S$, an $S$-linear quasigroup is a unital $S$-module, with automorphisms $\rho$ and $\lambda$ giving a (nonassociative) multiplication $x\cdot y=x^\rho+y^\lambda$. If $S$ is the field of complex numbers, then ordinary characters provide a complete linear isomorphism invariant for finite-dimensional $S$-linear quasigroups. Over other rings, it is an open problem to determine tractably computable isomorphism invariants. The paper investigates this isomorphism problem for $\mathbb{Z}$-linear quasigroups. We consider the extent to which ordinary characters classify $\mathbb{Z}$-linear quasigroups and their representations of the free group on two generators. We exhibit non-isomorphic $\mathbb{Z}$-linear quasigroups with the same ordinary character. For a subclass of $\mathbb{Z}$-linear quasigroups, equivalences of the corresponding ordinary representations are realized by permutational intertwinings. This leads to a new equivalence relation on $\mathbb{Z}$-linear quasigroups, namely permutational similarity. Like the earlier concept of central isotopy, permutational similarity is intermediate between isomorphism and isotopy.