Separation of variables for Uq(sln+1)+

Let Uq(sln+1)+ be the positive part of the quantized enveloping algebra Uq(sln+1). Using results of Alev-Dumas and Caldero related to the center of Uq(sl n+1)+, we show that this algebra is free over its center. This is reminiscent of Kostant's separation of variables for the enveloping algebra U(g) of a complex semisimple Lie algebra g, and also of an analogous result of Joseph-Letzter for the quantum algebra q(g). Of greater importance to its representation theory is the fact that U q(sln+1)+ is free over a larger polynomial subalgebra N in n variables. Induction from N to Uq(sl n+1)+ provides infinite-dimensional modules with good properties, including a grading that is inherited by submodules. (c) Canadian Mathematical Society 2005