Mickelsson algebras and inverse Shapovalov form

TL;DR

This paper introduces a method to construct reduction algebras associated with semi-simple Lie algebras using the inverse Shapovalov form, expanding algebraic tools in representation theory.

Contribution

It develops a novel construction of reduction algebras for pairs involving associative algebras and universal enveloping algebras using the inverse Shapovalov form.

Findings

Construction of the reduction algebra via inverse Shapovalov form

Application to classical and quantum universal enveloping algebras

Enhanced understanding of algebraic structures in Lie theory

Abstract

Let $\mathcal{A}$ be an associative algebra containing the classical or quantum universal enveloping algebra $U$ of a semi-simple complex Lie algebra. Let $\mathcal{J}\subset \mathcal{A}$ designate the left ideal generated by positive root vectors in $U$. We construct the reduction algebra of the pair $(\mathcal{A},\mathcal{J})$ via the inverse Shapovalov form of $U$.