Haar-Type Measures on Topological Quasigroups and Kunen's Theorem
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866912959889932288 |
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| author | Inoué, Takao |
| author_facet | Inoué, Takao |
| contents | Haar measure is a fundamental structure in harmonic analysis on locally compact groups. Its existence reflects the compatibility between topology and the associative algebraic structure of groups. In this paper we propose a framework for Haar-type measures on topological quasigroups. Since associativity is absent, strict translation invariance is generally too strong to expect. We therefore introduce quasi-invariant measures whose defect is measured by a modular cocycle attached to translations.
We then explain, in a detailed and cautious form, how Moufang-type identities may impose strong constraints on this cocycle. In particular, under additional quasi-invariance assumptions for right translations, the Moufang-type identity $(N1)$ leads naturally to a multiplicativity relation for the cocycle. This suggests a measure-theoretic interpretation of Kunen's theorem: the emergence of loop structure may be viewed as the collapse of a modular defect in the translation geometry of a quasigroup. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_06174 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Haar-Type Measures on Topological Quasigroups and Kunen's Theorem Inoué, Takao Group Theory Functional Analysis 20N05, 22A05, 28C10, 43A05 Haar measure is a fundamental structure in harmonic analysis on locally compact groups. Its existence reflects the compatibility between topology and the associative algebraic structure of groups. In this paper we propose a framework for Haar-type measures on topological quasigroups. Since associativity is absent, strict translation invariance is generally too strong to expect. We therefore introduce quasi-invariant measures whose defect is measured by a modular cocycle attached to translations. We then explain, in a detailed and cautious form, how Moufang-type identities may impose strong constraints on this cocycle. In particular, under additional quasi-invariance assumptions for right translations, the Moufang-type identity $(N1)$ leads naturally to a multiplicativity relation for the cocycle. This suggests a measure-theoretic interpretation of Kunen's theorem: the emergence of loop structure may be viewed as the collapse of a modular defect in the translation geometry of a quasigroup. |
| title | Haar-Type Measures on Topological Quasigroups and Kunen's Theorem |
| topic | Group Theory Functional Analysis 20N05, 22A05, 28C10, 43A05 |
| url | https://arxiv.org/abs/2603.06174 |