# Involutive representations of coordinate algebras and quantum spaces

## TL;DR

This paper develops covariant involutive representations of $rak{su}(2)$-related quantum space coordinate algebras, linking them to the Kontsevich product and proposing noncommutative scalar field theories free from UV/IR mixing.

## Contribution

It introduces $SO(3)$-covariant poly-differential involutive representations for $rak{su}(2)$-noncommutative spaces and connects them to the Kontsevich product, extending to semi-simple Lie groups.

## Key findings

- Representation of $rak{su}(2)$ coordinate operators via covariant poly-differential involutive maps.
- Quantized plane waves determined by polar decomposition and Wigner theorem constraints.
- Star-product equivalent to the Kontsevich product for the dual Poisson manifold.

## Abstract

We show that $\frak{su}(2)$ Lie algebras of coordinate operators related to quantum spaces with $\frak{su}(2)$ noncommutativity can be conveniently represented by $SO(3)$-covariant poly-differential involutive representations. We show that the quantized plane waves obtained 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 $\mathfrak{su}(2)$. 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.