Parabolic Stabilization and Cutting for Reduction Superalgebras

Abstract

The diagonal reduction algebra of a reductive Lie algebra $\mathfrak{g}$ is a localization of the Mickelsson algebra associated to the symmetric pair $(\mathfrak{g}\times \mathfrak{g},\mathfrak{g})$. In 2010, Khoroshkin and Ogievetsky introduced the methods of stabilization and cutting, which relate the commutation relations in the diagonal reduction algebra of ${\mathfrak{gl}}_m\oplus {\mathfrak{gl}}_n$ with those in the diagonal reduction algebra of ${\mathfrak{gl}}_{m+n}$. We extend this method to a wide range of reduction algebras, including all diagonal and differential reduction algebras for basic classical Lie superalgebras. We show how the method can be used for computing relations in the diagonal reduction algebra of ${\mathfrak{so}}_8$ and differential reduction algebra of ${\mathfrak{sp}}_{2n}$.