Gauge transformations on locally trivial quantum principal fibre bundles

We consider in this paper gauge transformations on locally trivial quantum principal fibre bundles (QPFB). If P , B, H , are the algebras of the total space, the base space, the structure group of the bundle, left (right) gauge transformations are defined as automorphisms of the left (right) B-module P which are adapted to the coaction of the Hopf algebra H and to the covering related to the local trivializations. Connections on the QPFB are in general not transformed into connections. For covariant connections, there are analogues of the classical formulas relating the connection and its gauge transform. This paper is a follow-up of [3]–[5]. We freely use the results of these papers. In [2], gauge transformations are defined as “vertical automorphisms” of the bundle. We start here with the more general definition of [1] (given there only for trivial bundles) and [7] of gauge transformations as convolution invertible linear maps from the Hopf algebra H to the total space algebra P (taken now in a different context) and consider the action of such transformations on differential geometric objects (connections and covariant derivatives). We define left (right) gauge transformations on a locally trivial QPFB in the sense of [2] and [4] as automorphisms of the left (right) B-module P which are compatible with the right coaction on the total space and with the local trivializations. On a trivial bundle, such gauge transformations correspond to convolution invertible (left case) and twisted convolution invertible (right case) linear maps from H to B. On a locally trivial QPFB, the corresponding objects are linear maps from H to P with these properties. If the structure group is a compact quantum group, gauge transformations can be characterized locally by elements of the algebras defined locally by the covering of the basis. Gauge transformations can also be defined on locally trivial quantum vector bundles (QVB) as module automorphism respecting the local trivializations. If a QVB is associated to some QPFB, every gauge transformation of the QPFB induces a gauge transformation on the QVB. Every gauge transformation on a QPFB induces a module isomorphism of the module of horizontal forms belonging to a differential structure of P (the module structure being with respect to the maximal embeddable LC differential algebra related to the differential structure). It follows that the set of covariant derivatives is invariant under gauge transformations. For the set of connections, this is true at least in the following two cases: If the differential structure supported by Deutsche Forschungsgemeinschaft, e-mail [email protected] supported by Sächsisches Staatsministerium für Wissenschaft und Kunst, e-mail [email protected] or [email protected]