Multiplier Hopf algebras of discrete type
New York, N.Y.
Journal of algebra. - New York, N.Y.
, p. 400-417
University of Antwerp
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, Delta) where A is an algebra over C, possibly without identity, and a is a comultiplication on A (a homomorphism of A into the multiplier algebra M(A x A) of A x A) satisfying certain properties. The typical example is the algebra A of complex functions with finite support in a group G, with pointwise multiplication and where the comultiplication is defined by (Delta f)(p, q) = f(pq) whenever f is an element of A and p, q is an element of G. A left cointegral in a multiplier Hopf algebra is an element h is an element of A such that ah = epsilon(a)h for all a is an element of A where epsilon is the counit of A. 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 cp on A such that (iota x phi)Delta(a) = phi(a)1 where iota is the identity map (and where the equation is to be considered in M(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 algebra A for a multiplier Hopf algebra (A, Delta) to have cointegrals (i.e., to be of discrete type). The algebras turn out to be Frobenius, quasi-Frobenius, and Kasch. (C) 1999 Academic Press.