Multiplier Hopf Algebras of Discrete Type

Abstract

In this paper, we study regular multiplier Hopf algebras with cointegrals. They are a certain class of multiplier Hopf algebras, still sharing many nice properties with the (much smaller class of) finite-dimensional Hopf algebras. Recall that a multiplier Hopf algebra is a pair (A,Δ) whereAis an algebra over C, possibly without identity, and Δ is a comultiplication onA(a homomorphism ofAinto the multiplier algebraM(A⊗A) ofA⊗A) satisfying certain properties. The typical example is the algebraAof complex functions with finite support in a groupG, with pointwise multiplication and where the comultiplication is defined by (Δf)(p,q)=f(pq) wheneverf∈Aandp,q∈G. A left cointegral in a multiplier Hopf algebra is an elementh∈Asuch thatah=ϵ(a)hfor alla∈Awhere ϵ is the counit ofA. In the group example, this is the function that is 1 on the identity of the group and 0 everywhere else. In this paper, we show that cointegrals are unique (up to a constant) if they exist and that they are faithful. We also show that on a regular multiplier Hopf algebra with a left cointegral, there exists also a left integral. Recall that a left integral is a linear functional ϕ onAsuch that (ι⊗ϕ)Δ(a)=ϕ(a)1 where ι is the identity map (and where the equation is to be considered inM(A)). A multiplier Hopf algebra with cointegrals is therefore an algebraic quantum group of discrete type. We will also obtain different necessary and sufficient conditions on the algebraAfor a multiplier Hopf algebra (A,Δ) to have cointegrals (i.e., to be of discrete type). The algebras turn out to be Frobenius, quasi-Frobenius, and Kasch.