Algebraic quantum groups and duality I

TL;DR

The paper addresses duality between a finite-dimensional Hopf algebra $A$ and its dual $\widehat{A}$, encoded by the canonical element $V\in B\otimes A$ satisfying $\langle V,a\otimes b\rangle=\langle a,b\rangle$. It extends this duality to the setting of regular multiplier Hopf algebras with integrals (algebraic quantum groups), where $V\in M(B\otimes A)$ and the Heisenberg algebra $C$ provides a canonical action that realises the duality as a map $a\otimes a'\mapsto \Delta(a)(1\otimes a')$ on $A\otimes A$. The work develops a broad suite of formulas relating the structures on $A$ and its dual $\widehat{A}$, including the Fourier transform $\mathcal{F}$ and transformations of modular data (such as the modular element $\delta$ and modular automorphisms $\sigma,\sigma'$), thereby unifying finite- and infinite-dimensional cases. This concise, pairing-based treatment lays the groundwork for Parts II and III, which treat multiplier Hopf $^*$-algebras with positive integrals and related positivity properties, respectively, and clarifies duality concepts for algebraic quantum groups and their operator-algebraic counterparts.

Abstract

Let $(A,Δ)$ be a finite-dimensional Hopf algebra. The linear dual $B$ of $A$ is again a finite-dimensional Hopf algebra. The duality is given by an element $V\in B\otimes A$, defined by $\langle V,a\otimes b\rangle=\langle a,b\rangle$ where $a\in A$ and $b\in B$. We use $\langle\,\cdot\, , \,\cdot\,\rangle$ for the pairings. In the introduction of this paper, we recall the various properties of this element $V$ as sitting in the algebra $B\otimes A$. More generally, we can consider an algebraic quantum group $(A,Δ)$. We use the term here for a regular multiplier Hopf algebra with integrals. For $B$ we now take the dual $\widehat A$ of $A$. It is again an algebraic quantum group. In this case, the duality gives rise to an element $V$ in the multiplier algebra $M(B\otimes A)$. Still, most of the properties of $V$ in the finite-dimensional case are true in this more general setting. The focus in this paper lies on various aspects of the duality between $A$ and its dual $\widehat A$. Among other things we include a number of formulas relating the objects associated with an algebraic quantum group and its dual. This note is meant to give a comprehensive, yet concise (and sometimes simpler) account of these known results. This is part I of a series of three papers on this subject. The case of a multiplier Hopf $^*$-algebra with positive integrals is treated in detail in part II and part III.