Braided symmetric algebras of simple Uq(sl2)-modules and their geometry

In the present paper we prove decomposition formulae for the braided symmetric powers of simple Uq(sl2)-modules, natural quantum analogues of the classical symmetric powers of a module over a complex semisimple Lie algebra. We show that their point modules form natural non-commutative curves and surfaces and conjecture that braided symmetric algebras give rise to an interesting non-commutative geometry, which can be viewed as a flat deformation of the geometry associated to their classical limits.