There are two notions of Yang-Mills action functional in noncommutative geometry. We show that for noncommutative n-torus both these notions agree. We also prove a structure theorem on the Hermitian structure of a finitely generated projective modules over spectrally invariant subalgebras of C -algebras.

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In this article, we obtain a complete list of inequivalent irreducible representations of the compact quantum group U q (2) for non-zero complex deformation parameters q, which are not roots of unity. The matrix coefficients of these representations are described in terms of the little q-Jacobi polynomials. The Haar state is shown to be faithful and an orthonormal basis of L 2 (U q (2)) is obtained. Thus, we have an explicit description of the Peter-Weyl decomposition of U q (2). As an application, we discuss the Fourier transform and establish the Plancherel formula. We also describe the decomposition of the tensor product of two irreducible representations into irreducible components. Finally, we classify the compact quantum groups U q (2).

We consider the algebra k[X 2 , XY, Y 2 ] where characteristic of the field k is zero. We compute a differential calculus, introduced earlier by the authors, by associating an algebraic spectral triple with this algebra. This algebra can also be viewed as the coordinate ring of the singular variety UV − W 2 and hence, is a quadratic algebra. We associate two canonical algebraic spectral triples with this algebra and its quadratic dual, and compute the associated Connes' calculus. We observe that the resulting Connes' calculi are also quadratic algebras, and they turn out to be quadratic dual to each other.

a b s t r a c tGiven a spectral triple (A, H, D) Connes associated a canonical differential graded algebra Ω • D (A). However, so far this has been computed for very few special cases. We identify suitable hypotheses on a spectral triple that helps one to compute the associated Connes' calculus for its quantum double suspension. This allows one to compute Ω • D for spectral triples obtained by iterated quantum double suspension of the spectral triple associated with a first order differential operator on a compact smooth manifold. This gives the first systematic computation of Connes' calculus for a large family of spectral triples.