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The differential algebra on the fuzzy sphere is constructed by applying Connes' scheme. The U (1) gauge theory on the fuzzy sphere based on this differential algebra is defined. The local U (1) gauge transformation on the fuzzy sphere is identified with the left U (N + 1) transformation of the field, where a field is a bimodule over the quantized algebra A N . The interaction with a complex scalar field is also given.

We examine the properties of the quantum Lorentz group SO q(3, 1) using the [Formula: see text] matrix given in Ref. 14. We show that this matrix together with the q-deformed metric C provide a representation of a BWM algebra. Using the projection operators which decompose the [Formula: see text] matrix into irreducible components, we give the general definition of the corresponding quantum space, i.e. the q-deformed Minkowski space and the q-deformed Clifford algebra. We also construct the q analog of Dirac matrices and show that they form a matrix representation of the q-deformed Clifford algebra.

Abstract:We give a derivation of the Dirac operator on the noncommutative 2-sphere within the framework of the bosonic fuzzy sphere and define Connes' triple. It turns out that there are two different types of spectra of the Dirac operator and correspondingly there are two classes of quantized algebras. As a result we obtain a new restriction on the Planck constant in Berezin's quantization. The map to the local frame in noncommutative geometry is also discussed.

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