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

We introduce an integral form image of the quantized enveloping algebra of sl2. The algebra image is exactly large enough so that the quasi-R-matrix is contained in a completion of image. We study several completions of the algebra image, and determine their centers. This study is motivated by a study of integrality properties of the quantum sl2 invariants of links and integral homology spheres.