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Published online by Cambridge University Press: 20 November 2018
We classify the affine actions of ${{U}_{q}}\left( sl\left( 2 \right) \right)$ on commutative polynomial rings in
$m\,\ge \,1$ variables. We show that, up to scalar multiplication, there are two possible actions. In addition, for each action, the subring of invariants is a polynomial ring in either
$m$ or
$m\,-\,1$ variables, depending upon whether
$q$ is or is not a root of 1.