Isomorphism invariants for linear quasigroups

Abstract

For a unital ring S, an S-linear quasigroup is a unital S-module, with automorphisms ρ and λ giving a (nonassociative) multiplication x • y = x ρ + y λ. 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 Z-linear quasigroups. We consider the extent to which ordinary characters classify Z-linear quasigroups and their representations of the free group on two generators. We exhibit non-isomorphic Z-linear quasigroups with the same ordinary character. For a subclass of Z-linear quasigroups, equivalences of the corresponding ordinary representations are realized by permutational intertwinings. This leads to a new equivalence relation on Z-linear quasigroups, namely permutational similarity. Like the earlier concept of central isotopy, permutational similarity is intermediate between isomorphism and isotopy.