Involutive representations of coordinate algebras and quantum spaces

Abstract

We show that mathfrak{s}mathfrak{u}(2) Lie algebras of coordinate operators related to quantum spaces with mathfrak{s}mathfrak{u}(2) noncommutativity can be conveniently represented by SO(3)-covariant poly-differential involutive representations. We show that the quantized plane waves ob-tained from the quantization map action on the usual exponential functions are determined by polar decomposition of operators combined with constraint stemming from the Wigner theorem for SU(2). Selecting a subfamily of ∗-representations, we show that the resulting star-product is equivalent to the Kontsevich product for the Poisson manifold dual to the finite dimensional Lie algebra mathfrak{s}mathfrak{u}(2) . We discuss the results, indicating a way to extend the construction to any semi-simple non simply connected Lie group and present noncommutative scalar field theories which are free from perturbative UV/IR mixing.