arXiv:q-alg/9602022v1 12 Feb 1996 DAMTP/95-74 COALGEBRA GAUGE THEORY 1 Tomasz Brzezi´ nski and Shahn Majid 2 Department of Applied Mathematics & Theoretical Physics University of Cambridge, Cambridge CB3 9EW December 1995 – revised January 1996 Abstract We develop a generalised gauge theory in which the role of gauge group is played by a coalgebra and the role of principal bundle by an algebra. The theory provides a unifying point of view which includes quantum group gauge theory, embeddable quantum homogeneous spaces and braided group gauge theory, the latter being introduced now by these means. Examples include ones in which the gauge groups are the braided line and the quantum plane. Keywords: gauge theory – coalgebra – quantum group – braided group – quan- tum plane – connection – bicrossproduct – factorisation 1 Introduction In a recent paper [Brz95] it was shown by the first author that a generalisation of the quantum group principal bundles introduced in [BM93] is needed if one wants to include certain ‘embeddable’ quantum homogeneous spaces, such as the full family of quantum two-spheres of Podle´ s [Pod87]. A one-parameter specialisation of this family was used in [BM93] in construction of the q -monopole, but the general members of the family do not have the required canonical fibering. The required generalised notion of quantum principal bundles proposed in [Brz95], also termed a C -Galois extension, consists of an algebra P , a coalgebra C with a distinguished element e and a right action of P on P ⊗ C satisfying certain conditions. In the present paper we develop a version of such ‘coalgebra principal bundles’ based on a map ψ : C ⊗ P → P ⊗ C and e ∈ C , and giving now a theory of connections on them. Another motivation for the paper is the search for a generalisation of gauge theory powerful enough to include braided groups[Maj91][Maj93b][Maj93a] as the gauge group. 1 Research supported by the EPSRC grant GR/K02244 2 Royal Society University Research Fellow and Fellow of PembrokeCollege, Cambridge. On leave 1995 + 1996 at the Department of Mathematics, Harvard University, Cambridge MA02138, USA 1