Reflections on the Larson-Sweedler theorem for (weak) multiplier Hopf algebras

TL;DR

This paper revisits the Larson-Sweedler theorem, showing that for multiplier and weak multiplier Hopf algebras, the existence of a counit can be derived from other conditions, simplifying the theoretical framework.

Contribution

It demonstrates that the fullness condition for coproducts in multiplier Hopf algebras is unnecessary, as it follows from other assumptions, thus generalizing the Larson-Sweedler theorem.

Findings

The fullness condition is redundant in multiplier Hopf algebras.

The existence of a counit follows from other properties in these algebras.

Implications for the theory of locally compact quantum groups.

Abstract

Let $A$ be an algebra with identity and $\Delta:A\to A\otimes A$ a coproduct that admits a counit. If there exist a faithful left integral and a faithful right integral, one can construct an antipode and $(A,\Delta)$ is a Hopf algebra. This is the Larson-Sweedler theorem. There are generalizations of this result for multiplier Hopf algebras, weak Hopf algebras and weak multiplier Hopf algebras. In the case of a multiplier Hopf algebra, the existence of a counit can be weakened and can be replaced by the requirement that the coproduct is full. A similar result is true for weak multiplier Hopf algebras. What we show in this note is that in fact the result for multiplier Hopf algebras can still be obtained without the condition of fullness of the coproduct. As it turns out, this property will already follow from the other conditions. Consequently, also in the original theorem for Hopf algebras, the existence of a counit is a consequence of the other conditions. This slightly generalizes the original result. The situation for weak multiplier Hopf algebras seems to be more subtle. We discuss the problems and see what is still possible here. We consider these results in connection with the development of the theory of locally compact quantum groups. This is discussed in an appendix.