Noncommutative geometry of Homogenized Quantum Sl(2,)

We examine the relationship between certain noncommutative analogues of projective 3-space, ³, and the quantized enveloping algebras U_q (sl₂). The relationship is mediated by certain noncommutative graded algebras S, one for each q ∈ ×, having a degree-two central element c such that S[c^-1]₀ ≅ U_q (sl₂). The noncommutative analogues of ³ are the spaces Proj_nc(S). We show how the points, fat points, lines, and quadrics, in Proj_nc(S), and their incidence relations, correspond to finite-dimensional irreducible representations of U_q (sl₂), Verma modules, annihilators of Verma modules, and homomorphisms between them.