Linear aspects of equational triality in quasigroups

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2020-01-01
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Nowak, Alex
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Jonathan D. H. Smith
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Mathematics
Abstract

We approach quasigroup triality from two vantages: $Q$-modules and quantum quasigroups. The former requires considerable exposition and a review of the Fundamental Theorems of Quasigroup Modules due to Smith. There are six varieties associated with triality symmetry. We give presentations for rings whose module theories coincide with the Beck modules for quasigroups in each of these six varieties, plus the idempotent extensions of each of these varieties. In three of the idempotent cases, we are able to give abstract structure theorems which present these rings as free products of PIDs and free group rings.

We define a Mendelsohn triple system (MTS) of order coprime with $3$, and having multiplication affine over an abelian group, to be \emph{affine non-ramified}. We classify, up to isomorphism, all affine non-ramified MTS and enumerate isomorphism classes (extending the work of Donovan, Griggs, McCourt, Opr\v{s}al, and Stanovsk\'{y}). As a consequence, all entropic MTS and distributive MTS of order coprime with $3$ are classified. The classification is accomplished via the representation theory of the Eisenstein integers, $\mathbb{Z}[\zeta]=\mathbb{Z}[X]/(X^2-X+1)$. Partial results on the isomorphism problem for affine MTS with order divisible by $3$ are given, and a complete classification is conjectured. We also prove that for any affine MTS, the qualities of being non-ramified, pure, and self-orthogonal are equivalent.

We conclude with an exploration of conjugates and triality in quantum quasigroups, which offer a nonassociative generalization of Hopf algebras. We show that if $Q$ is a finite quasigroup and $G\leq \text{Aut}(Q)$, then the Hopf smash product $KG\#K(Q)$ exhibits a quantum quasigroup structure which, despite being noncoassociative and noncocommutative, has a set of conjugates on which $S_3$ acts.

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Fri May 01 00:00:00 UTC 2020