A duality theorem for Hopf module algebras

Crossed Products and Inner Actions of Hopf Algebras

“…We then consider various natural actions of H or H° on A # H, or H° # H, and determine conditions under which they are inner or outer. In particular it is shown that the RL-condition of Blattner and Montgomery [2] can be reformulated in terms of whether a certain action is inner. Much of the work in this section can be best viewed in terms of nonabelian cohomology of Hopf algebras; however this will be discussed in a subsequent paper.…”

“…(1) the weak action is inner, (2) there exists an invertible u G i\omk(H, A) such that Then u is convolution invertible in Homfc(//, A), u~x = u o S (so in particular u~x is an algebra antihomomorphism), and the weak action implemented by u is in fact an action.…”

“…The converse is false. If kG is residually finite dimensional (i.e., G is residually linear [23]), then a (fcG)°-module algebra A is graded in such a way that the coaction p of kG gives rise to the original (fcG)°-action if and only if that action is fcG-locally finite (see [2,Corollary 4.9]). If G is finite, the (fcG)*-action is given in terms of the grading by (3.1) f-a=YUix)ax xeG for o G A, /G(fcG)*.…”

“…Let ipu denote the inner weak action of H on A implemented by it (Lemma 1.4). Lemma 1.13 (2) shows that all inner weak actions are of the form ipu for some such u. Let ipf denote the trivial action.…”

Rings Graded by Polycyclic-by-Finite Groups

“…We comment that the duality theorem of Cohen and Montgomery has been extended by R. Blattner and S. Montgomery [4] to handle certain infinite groups as a corollary to a more general theorem on Hopf algebra actions. Other studies of the duality have been made by M. van den Bergh [2] and J. Osterburg [10].…”