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Modular Cocycles and Haar-Type Measures on Topological Loops

Takao Inoué

Abstract

This paper is a continuation of the author's companion work \cite{InoueQuasi} on Haar-type measures for topological quasigroups, where the quasigroup setting was analyzed in connection with Kunen's theorem. We extend that framework to locally compact topological loops and study Haar-type (quasi-invariant) Radon measures together with modular cocycles describing the distortion of such measures under translations. Unlike the classical group case, the composition of translations in a loop is affected by the failure of associativity, which produces an additional correction term governed by an associativity deviation map. We derive the resulting cocycle relation and show that loop identities, in particular Moufang- and Kunen-type identities, impose structural restrictions on the modular data. In the associative limit, the cocycle reduces to the classical modular function of a locally compact group.

Modular Cocycles and Haar-Type Measures on Topological Loops

Abstract

This paper is a continuation of the author's companion work \cite{InoueQuasi} on Haar-type measures for topological quasigroups, where the quasigroup setting was analyzed in connection with Kunen's theorem. We extend that framework to locally compact topological loops and study Haar-type (quasi-invariant) Radon measures together with modular cocycles describing the distortion of such measures under translations. Unlike the classical group case, the composition of translations in a loop is affected by the failure of associativity, which produces an additional correction term governed by an associativity deviation map. We derive the resulting cocycle relation and show that loop identities, in particular Moufang- and Kunen-type identities, impose structural restrictions on the modular data. In the associative limit, the cocycle reduces to the classical modular function of a locally compact group.
Paper Structure (15 sections, 8 theorems, 40 equations, 2 figures)

This paper contains 15 sections, 8 theorems, 40 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Topological Loops and Translation Operators
  3. Haar-Type Measures and Modular Cocycles
  4. Deviation of Translation Composition
  5. Deviation homeomorphisms
  6. Measure-class correction
  7. Loop Identities and Constraints on the Cocycle
  8. A general rigidity principle
  9. Moufang-type constraints
  10. The Kunen identity
  11. Interpretation
  12. A Basic Example
  13. Conclusion and Future Directions
  14. Structural Overview of the Modular Cocycle Framework
  15. Structural Diagram and Relation to the Quasigroup Framework

Key Result

Proposition 2.4

For every $a\in L$, the maps $L_a$ and $R_a$ are bijections of $L$. If $L$ is a topological loop, they are homeomorphisms.

Figures (2)

  • Figure 1: Conceptual structure of the modular cocycle framework for Haar-type measures on topological loops. The failure of associativity produces an associativity deviation, which contributes a correction term to the cocycle relation. Loop identities such as Moufang- and Kunen-type identities then impose structural restrictions on the resulting modular data.
  • Figure 2: Comparison between the quasigroup framework of InoueQuasi and the present loop-theoretic extension. The upper row summarizes the quasigroup setting, where Haar-type measures were studied in relation to Kunen's theorem. The lower part summarizes the present paper: Haar-type measures on locally compact topological loops give rise to Radon--Nikodym data and modular cocycles, with an additional correction term produced by associativity deviation. Moufang- and Kunen-type identities then impose structural restrictions on the modular data, and in the associative limit the theory recovers the classical modular function.

Theorems & Definitions (28)

  • Remark 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Definition 3.3
  • ...and 18 more