The nonstandard quantum plane

Let Uq be the quantum group associated to sl2(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 kq[x,y], considered as an L(Uq)-structure, where L(Uq) is the language of representations of Uq. It is proved that the lattice of definable k-subspaces of kq[x,y] is complemented. This is deduced from the same result for the Uq-module M, which is defined to be the direct sum of all finite dimensional representations of Uq. It follows that the ring of definable scalars for the quantum plane is a von Neumann regular epimorphic ring extension of the quantum group Uq.