Rings of Fractions of Reduction Algebras

We establish the absence of zero divisors in the reduction algebra of a Lie algebra ${\mathfrak{g}}$ with respect to its reductive Lie subalgebra ${\mathfrak{k}}$ . We identify the field of fractions of the diagonal reduction algebra of ${\mathfrak{sl}}_2$ with the standard skew field; as a by-product we obtain a two-parametric family of realizations of this diagonal reduction algebra by differential operators. We also present a new proof of the Poincare–Birkhoff–Witt theorem for reduction algebras.