Reductions of piecewise trivial principal comodule algebras

Details Reductions of piecewise trivial principal comodule algebras Reductions of piecewise trivial principal comodule algebras

2010-12-31

arxiv.org/abs/1101.0201v2

Mathematics

Quantum Algebra

16 pages, proof of the main result included

Scientific paper

The structure group of a principal bundle is reducible to a subgroup if there exists a local trivialisation with respect to which all transition functions take values in this subgroup. Conversely, if a principal bundle is reducible to a locally trivial principal subbundle, then there exists a local trivialisation of the bundle such that all transition functions take values in the structure group of the subbundle. We prove a noncommutative-geometric counterpart of this theorem. To this end, we employ the concept of a piecewise trivial principal comodule algebra as a suitable replacement of a locally trivial compact principal bundle. To enclose natural and geometrically interesting noncommutative examples, we use smash products (cocycle-free crossed products) rather than tensor products as a generalisation of trivial principal bundles. These examples serve as a testing ground for our reduction theorem, and include bundles over noncommutative lens spaces.

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