Integral Dual of some infinite dimensional Hopf quasigroups

TL;DR

This paper constructs the integral dual of certain infinite dimensional Hopf quasigroups using faithful integrals, revealing that the dual has a structure akin to Hopf coquasigroups, thus extending the theory of quantum algebraic structures.

Contribution

It introduces a method to construct the integral dual of infinite dimensional Hopf quasigroups based on faithful integrals, showing the dual's structure parallels Hopf coquasigroups.

Findings

The integral dual of the considered Hopf quasigroups is a regular multiplier Hopf coquasigroup.

The integral dual possesses a faithful integral, similar to the original structure.

Existence and uniqueness of faithful integrals are established for these quasigroups.

Abstract

For an infinite dimensional Hopf quasigroup, if the faithful integral exists, then it is unique up to scalar. Base on the faithful integrals, we construct the integral dual of a class of infinite dimensional Hopf quasigroups, and show that the integral dual has a similar structure to Hopf coquasigroup, which is a regular multiplier Hopf coquasigroup with a faithful integral.