2023/04/26 by Van Daele, Alfons
#16T05 #FOS: Mathematics #Quantum Algebra (math.QA) #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.2304.13448
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∈ B⊗ A, defined by ⟨ V,a⊗ b⟩=⟨ a,b⟩ where a∈ A and b∈ B. We use ⟨ ⋅ , ⋅ ⟩ for the pairings. In the introduction of this paper, we recall the various properties of this element V as sitting in the algebra B⊗ 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⊗ 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.