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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2602.21690 |
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| _version_ | 1866912926271537152 |
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| author | Inoué, Takao |
| author_facet | Inoué, Takao |
| contents | We develop a theory of chirality and racemization on isotopy classes of finite loops, formulated intrinsically within the loop isotopy groupoid understood in the categorical sense. Motivated by earlier work on quasigroups \cite{InoueQuasiChirality} and by the classical medical paradigm of mirror-related enantiomers, we restrict admissible mirror transitions to those generated by intrinsic, unit-preserving symmetries. Within this framework, racemization is modeled as a two-state dynamics on isotopy classes, with an effective rate determined by the presence of mirror-isotopisms. Our main result shows that this rate vanishes if and only if no loop isotopism exists between a loop and its opposite, providing a structural criterion for chirality. A strengthened variant based on translation-generated symmetries is discussed in the appendix. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_21690 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Chirality and Racemization on Isotopy Classes of Loops: A Groupoid-Based Structural Theory Inoué, Takao Group Theory 20N05, 20N02 We develop a theory of chirality and racemization on isotopy classes of finite loops, formulated intrinsically within the loop isotopy groupoid understood in the categorical sense. Motivated by earlier work on quasigroups \cite{InoueQuasiChirality} and by the classical medical paradigm of mirror-related enantiomers, we restrict admissible mirror transitions to those generated by intrinsic, unit-preserving symmetries. Within this framework, racemization is modeled as a two-state dynamics on isotopy classes, with an effective rate determined by the presence of mirror-isotopisms. Our main result shows that this rate vanishes if and only if no loop isotopism exists between a loop and its opposite, providing a structural criterion for chirality. A strengthened variant based on translation-generated symmetries is discussed in the appendix. |
| title | Chirality and Racemization on Isotopy Classes of Loops: A Groupoid-Based Structural Theory |
| topic | Group Theory 20N05, 20N02 |
| url | https://arxiv.org/abs/2602.21690 |