On Modules Associated to Coalgebra Galois Extensions
Author: Montgomery T.S.
Source: Journal of Algebra, Volume 215, Number 1, May 1999 , pp. 290-317(28)
Publisher: Academic Press
Abstract:
For a given entwining structure (A, C)
involving an algebra A, a coalgebra C, and an entwining map
: C
A
A
C, a category MCA(
) of right (A, C)
-modules is defined and its structure analysed. In particular, the notion of a measuring of (A, C)
to (Ã, C~)
~ is introduced, and certain functors between MCA(
) and MC~Ã(
~) 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
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
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
is studied in the case when V is finite-dimensional and in the case when the canonical entwining map is bijective. Copyright 1999 Academic Press.
Language: English
Document Type: Research article

Click here for Page Help