Associated quantum vector bundles and symplectic structure on a quantum plane

Details Associated quantum vector bundles and symplectic structure on a quantum plane Associated quantum vector bundles and symplectic structure on a quantum plane

1999-08-06

arxiv.org/abs/math-ph/9908007v2

Czech. J. Phys. 50 (2000), 29-36

Physics

Mathematical Physics

10 pages, no figures, Latex; (not so compressed version of the) Contribution to the Proceedings of the 8th International Collo

Scientific paper

10.1023/A:1022860612069

We define a quantum generalization of the algebra of functions over an associated vector bundle of a principal bundle. Here the role of a quantum principal bundle is played by a Hopf-Galois extension. Smash products of an algebra times a Hopf algebra H are particular instances of these extensions, and in these cases we are able to define a differential calculus over their associated vector bundles without requiring the use of a (bicovariant) differential structure over H. Moreover, if H is coquasitriangular, it coacts naturally on the associated bundle, and the differential structure is covariant. We apply this construction to the case of the finite quotient of the SL_q(2) function Hopf algebra at a root of unity (q^3=1) as the structure group, and a reduced 2-dimensional quantum plane as both the "base manifold" and fibre, getting an algebra which generalizes the notion of classical phase space for this quantum space. We also build explicitly a differential complex for this phase space algebra, and find that levels 0 and 2 support a (co)representation of the quantum symplectic group. On this phase space we define vector fields, and with the help of the Sp_q structure we introduce a symplectic form relating 1-forms to vector fields. This leads naturally to the introduction of Poisson brackets, a necessary step to do "classical" mechanics on a quantum space, the quantum plane.

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