Title




Multiplier Hopf algebras of discrete type
 
Author




 
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) finitedimensional 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, quasiFrobenius, and Kasch. (C) 1999 Academic Press. 
 
Language




English
 
Source (journal)




Journal of algebra.  New York, N.Y., 1964, currens
 
Publication




New York, N.Y.
:
Academic Press
,
1999
 
ISSN




00218693
 
DOI




10.1006/JABR.1998.7717
 
Volume/pages




214
:2
(1999)
, p. 400417
 
ISI




000079685800002
 
Full text (Publisher's DOI)




 
Full text (open access)




 
