Abstract
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 Poincaré–Birkhoff–Witt theorem for reduction algebras.
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On leave of absence from P.N. Lebedev Physical Institute, Theoretical Department, Leninsky prospekt 53, 119991 Moscow, Russia.
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Khoroshkin, S., Ogievetsky, O. Rings of Fractions of Reduction Algebras. Algebr Represent Theor 17, 265–274 (2014). https://doi.org/10.1007/s10468-012-9397-4
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DOI: https://doi.org/10.1007/s10468-012-9397-4