The cointegral theory of weak multiplier Hopf algebras

TL;DR

This paper develops the theory of cointegrals in weak multiplier Hopf algebras, establishing conditions for integrals, and exploring properties like Frobenius structures and duality in algebraic quantum groupoids.

Contribution

It introduces cointegrals in weak multiplier Hopf algebras, characterizes when integrals exist, and links these structures to Frobenius properties and duality in algebraic quantum groupoids.

Findings

A discrete type algebra has a faithful set of cointegrals.

Single faithful cointegral implies Frobenius and quasi-Frobenius properties.

Dual of an algebraic quantum groupoid with a faithful cointegral is a weak Hopf algebra.

Abstract

In this paper, we introduce and study the notion of cointegrals in a weak multiplier Hopf algebras $(A, \Delta)$. A cointegral is a non-zero element $h$ in the multiplier algebra $M(A)$ such that $ah=\v_t(a)h$ for any $a\in A$. When $A$ has a faithful set of cointegrals (now we call $A$ of {\it discrete type}), we give a sufficient and necessary condition for existence of integrals on $A$. Then we consider a special case, i.e., $A$ has a single faithful cointegral, and we obtain more better results, such as $A$ is Frobenius, quasi-Frobenius, et al. Moreover when an algebraic quantum groupoid $A$ has a faithful cointegral, then the dual $\widehat{A}$ must be weak Hopf algebra. In the end, we investigate when $A$ has a cointegral and study relation between compact and discrete type.