Reduction Algebras Involving Symplectic Lie Algebras

Abstract

A complete presentation for the diagonal reduction algebra of $\mathfrak{gl}_n$ was found by Khoroshkin and Ogievetsky. In this paper we present a complete set of ordering relations for a reduction algebra $\mathcal{B}_n$ associated to $\mathfrak{gl}_n$ in a parabolic subalgebra of $\mathfrak{sp}_{2n}$. We also study its connection to the diagonal reduction algebra of $\mathfrak{sp}_{2n}$. We use braid group action by Zhelobenko automorphisms to reduce the computations required. Another important tool we employ is the ABRR equation for the extremal projector. As an application we prove that all homogeneous ordering relations for the diagonal reduction algebra of $\mathfrak{sp}_{2n}$ are obtained from the relations in $\mathcal{B}_n$ by applying type $C_n$ Zhelobenko automorphisms.