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Haar-Type Measures on Topological Quasigroups and Kunen's Theorem

Takao Inoué

Abstract

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.

Haar-Type Measures on Topological Quasigroups and Kunen's Theorem

Abstract

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 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.
Paper Structure (28 sections, 8 theorems, 136 equations)

This paper contains 28 sections, 8 theorems, 136 equations.

Table of Contents

  1. Introduction
  2. Topological Quasigroups and Translation Operators
  3. The Moufang Identity and Translation Calculus
  4. Haar-Type Measures and Modular Cocycles
  5. Quasi-invariance under left translations
  6. Two-sided quasi-invariance
  7. Elementary consequences
  8. The Moufang Identity and Translation Calculus
  9. Deriving Cocycle Relations from $(N1)$
  10. Representation of the translation group
  11. Group Case
  12. The $ax+b$ Group as a Concrete Model
  13. Left Haar measure
  14. Right translations and the modular function
  15. Existence Questions and the Role of the Translation Group
  16. ...and 13 more sections

Key Result

Theorem 1

Let $Q$ be a quasigroup. Then the Moufang identity holds for all $x,y,z\in Q$ if and only if the translation operators satisfy

Theorems & Definitions (37)

  • Remark 1
  • Definition 1
  • Definition 2
  • Remark 2
  • Definition 3
  • Theorem 1: Translation form of the Moufang identity (N1)
  • proof
  • Definition 4
  • Remark 3
  • Remark 4
  • ...and 27 more