Finiteness conditions for a hopf algebra with a nonzero integral

Abstract

In [l] Heyneman and the present author considered various finiteness conditions for a coalgebra and its linear dual algebra. The purpose of this paper is to specialize the ideas of [I] and [3, 41 to Hopf algebras with a nonzero left integral, and to study the antipode and one-dimensional ideals of the dual algebra. Suppose A is a Hopf algebra which has a nonzero left integral. Then if D CA is a finite-dimensional subcoalgebra, D(=) is finite-dimensional. In particular, A is locally finite. If the coradical A,, is a reflexive coalgebra, then A is right strongly reflexive (cofinite right ideals of A* are finitely generated). By the discussion of [ 1, Sect. 3.71 one can expect many Hopf algebras with a nonzero left integral over an infinite field to be reflexive-in particular, for many Hopf algebras with a nonzero left integral over an infinite field all finitedimensional right A*-modules are rational. Let Jr C A* (resp., ST) d enote the space of left (resp., right) integrals for A. If t: A + A is any injective bialgebra map then t*(Jr) = Jr. In particular s*(jJ = Jr where s is the antipode of A. Using this observation we prove that s is bijective if sL f (0). We d iscuss the connection between the one dimensional ideals of A* and the grouplikes of A. The conclusions we draw generalize some of the results of [S]. Let J-C A be an injective hull of k . 1 (as a left A-comodule). If s1 # (0) then we show JA, = A. If A,, is a Hopf subalgebra in addition then A = A, for some n (thus Rad A* is nilpotent). This paper could easily be formulated in terms of the theory of injectives in the category of right A-comodules, but we choose to use basic properties of idempotents for the sake of completeness.