A duality theorem for crossed products of Hopf algebras

Abstract

Several variations have appeared recently, see [l-5, 8, 101. Among the best are [2, 2.1; 3,2.2], but the former is restricted to the case of a trivial cocycle 0 and the latter to the case of finite-dimensional H. The main purpose of this paper is to prove a common generalization of these two theorems. In Section 2 we first recall preliminaries about crossed products, then we introduce some new algebra constructions, and we prove related, useful results. In particular, we define an algebra A #, H #Op U that will replace (A #, H) # U in the main theorem, and we show a relationship between the two algebras. In the new construction U is allowed to be more general; for example, we may take U= H*. Section 3 introduces a subalgebra H” of H* with a comodule structure co: H” -+ H@ H”. They are our tools to formulate and handle certain awkward assumptions in the main theorem; these can be regarded as finiteness conditions, and they correspond to the U-local finiteness and RL-condition of [2]. Section 4 contains the duality theorem and its proof. In Section 5 we draw some corollaries. In particular, we show that [2, 2.1; 3,2.2] do follow from our theorem. 153 0021-8693/92 $3.00