Mickelsson algebras and inverse Shapovalov form

TL;DR

This work develops a unified framework for Mickelsson (reduction) algebras $Z(\mathcal{A},\mathfrak{g})$ by embedding them into localized reductions $\hat{Z}(\mathcal{A},\mathfrak{g})$ and linking them to inverse and standard Shapovalov form data. It introduces two complementary constructions based on right and left Shapovalov matrices, mediated by two comultiplications $\Delta$ and $\tilde{\Delta}$ connected through the R-matrix, and employs Hasse-diagram routes to produce explicit PBW-type descriptions. A central theme is the interplay between Mickelsson generators, invariant (or quasi-invariant) tensors, and matrix factorizations of Shapovalov data, leading to explicit realizations of $\hat{Z}(\mathcal{A},\mathfrak{g})$ in both classical and quantum settings. The left- and right- Shapovalov perspectives are shown to be mutually informative, with quantum Lax operators providing a concrete realization for reductive pairs. Overall, the paper enhances the toolbox for restricting representations and constructing explicit bases in Mickelsson/reduction algebras via Shapovalov-theoretic methods.

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 $U$.

Theorems & Definitions (21)

• Definition 3.1
• Theorem 3.2: cf. K
• Proposition 4.1
• proof
• Proposition 4.2: MS
• Example 4.3
• Lemma 5.1
• proof
• Proposition 5.2
• proof