Classical Principal Fibre Bundles from a Quantum Group Viewpoint

Abstract

In this short article we review how the classical theory of principal fibre bundles (PFB) transcribes in an algebraic formalism. In this dual formulation, a PFB is given by a right co-module algebra $\mathcal{P}$ over a Hopf algebra $\mathcal{H}$ with a mapping ${\Delta }_R:\mathcal{P}\to \mathcal{P}\otimes \mathcal{H}$. In our case $\mathcal{P}$ is the (commutative) C*-algebra of complex-valued continuous functions on the total space P and $\mathcal{H}$ is the Hopf algebra of complex-valued functions on the structure group G. These underlying spaces are endowed with a topology only. The subalgebra $\mathcal{B}$ of ${\Delta }_R$-invariant elements is identified with the algebra of complex-valued functions on the base space B. In order to define horizontal one-forms, a differential calculus is needed. Since no a priori differential structure is assumed, we use the calculus of the universal differential envelope ${\Omega }^{\bullet }(\mathcal{P})$ which can be defined on any unital algebra. A connection on the PFB is then defined by a splitting of the universal one-forms as a direct sum of horizontal and vertical subspaces : ${\Omega }^1(\mathcal{P})={\Gamma }_{hor}\oplus {\Gamma }_{ver}$. In case of a strong connection in a trivial PFB, the general expression and gauge transformation of the connection one-form and the curvature two-form are given. A locally trivial PFB can be constructed through a gluing procedure of a cover of the algebra $\mathcal{P}$ (see this meeting's poster session P112, where examples are given).

arXiv comment8 pages, communication at the XIVth Brazilian Meeting on Particles and Fields, october 2003

References in corpus (1)

• Connections on locally trivial quantum principal fibre bundles