On Modules Associated to Coalgebra Galois Extensions

For a given entwining structure (A,C)ψ involving an algebra A, a coalgebra C, and an entwining map ψ : C ⊗ A → A ⊗ C, a category MA(ψ) of right (A,C)ψmodules is defined and its structure analysed. In particular, the notion of a measuring of (A,C)ψ to (Ã, C̃)ψ̃ is introduced, and certain functors between M C A(ψ) and M Ã (ψ̃) induced by such a measuring are defined. It is shown that these functors are inverse equivalences iff they are exact (or one of them faithfully exact) and the measuring satisfies a certain Galois-type condition. Next, left modules E and right modules Ē associated to a C-Galois extension A of B are defined. These can be thought of as objects dual to fibre bundles with coalgebra C in the place of a structure group, and a fibre V . Cross-sections of such associated modules are defined as module maps E → B or Ē → B. It is shown that they can be identified with suitably equivariant maps from the fibre to A. Also, it is shown that a C-Galois extension is cleft if and only if A = B⊗C as left B-modules and right C-comodules. The relationship between the modules E and Ē is studied in the case when V is finite-dimensional and in the case when the canonical entwining map is bijective. 1991 Mathematics Subject Classification. Primary 16W30. Secondary 17B37, 81R50. Lloyd’s Tercentenary Fellow. On leave from: Department of Theoretical Physics, University of Lódź, Pomorska 149/153, 90–236 Lódź, Poland. e-mail: [email protected]