Modular Cocycles and Haar-Type Measures on Topological Loops

TL;DR

This paper extends Haar measure theory to locally compact topological loops, analyzing how non-associativity affects measure invariance and introduces modular cocycles, with structural restrictions from loop identities.

Contribution

It generalizes Haar-type measures to topological loops, incorporating modular cocycles and associativity deviations, expanding the classical group framework.

Findings

Derived cocycle relations for topological loops.

Identified structural restrictions from loop identities.

Connected non-associative measures to classical group theory.

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.