An integral form of the quantized enveloping algebra of sl_2 and its completions

We introduce an integral form U of the quantized enveloping algebra of sl_2. The algebra U is just large enough so that the quasi-R-matrix is contained in a completion of U\otimes U. We study several completions of the algebra U, and determine their centers. This study is motivated by a study of integrality properties of the quantum sl_2 invariants of links and integral homology spheres.