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

TL;DR

This paper introduces an integral form of the quantized enveloping algebra of sl_2, explores its completions, and analyzes their centers, motivated by quantum invariants of links and homology spheres.

Contribution

It defines a new integral form of the algebra and investigates its completions and centers, advancing understanding of quantum invariants and algebraic structures.

Findings

The algebra U contains the quasi-R-matrix within its completion.

Several completions of U are studied and characterized.

Centers of the completions are explicitly determined.

Abstract

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.