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.
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Jurić, T., Poulain, T. & Wallet, JC. Involutive representations of coordinate algebras and quantum spaces. J. High Energ. Phys. 2017, 116 (2017). https://doi.org/10.1007/JHEP07(2017)116
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DOI: https://doi.org/10.1007/JHEP07(2017)116