The nonstandard quantum plane

Let U q be the quantum group associated to s l 2 ( k ) with char ( k ) ≠ 2 and q ∈ k not a root of unity. The article is devoted to the model-theoretic study of the quantum plane k q [ x , y ] , considered as an L ( U q ) -structure, where L ( U q ) is the language of representations of U q . It is proved that the lattice of definable k -subspaces of k q [ x , y ] is complemented. This is deduced from the same result for the U q -module M , which is defined to be the direct sum of all finite dimensional representations of U q . It follows that the ring of definable scalars for the quantum plane is a von Neumann regular epimorphic ring extension of the quantum group U q .