The Nonstandard quantum plane

Let Uq be the quantum group associated to sl2.k with char.k \neq 2 and q not a root of unity. The article is devoted to the model-theoretic study of the quantum plane kqx[; 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 kqTx; yU 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.