Physics – Mathematical Physics
Scientific paper
2003-12-05
Physics
Mathematical Physics
8 pages, communication at the XIVth Brazilian Meeting on Particles and Fields, october 2003
Scientific paper
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 ${\cal P}$ over a Hopf algebra ${\cal H}$ with a mapping $\Delta_R:{\cal P}\to{\cal P}\otimes{\cal H}$. In our case ${\cal P}$ is the (commutative) C*-algebra of complex-valued continuous functions on the total space P and ${\cal H}$ is the Hopf algebra of complex-valued functions on the structure group G. These underlying spaces are endowed with a topology only. The subalgebra ${\cal B}$ 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 $\Omega^\bullet({\cal P})$ 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 : $\Omega^1({\cal P})=\Gamma_{hor}\oplus\Gamma_{ver}$. 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 ${\cal P}$ (see this meeting's poster session P112, where examples are given).
da Silva Amaro Rica
Sigaud C.
Vanhecke F. J.
No associations
LandOfFree
Classical Principal Fibre Bundles from a Quantum Group Viewpoint does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Classical Principal Fibre Bundles from a Quantum Group Viewpoint, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Classical Principal Fibre Bundles from a Quantum Group Viewpoint will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-474933