Regularization of Mickelsson generators for non-exceptional quantum groups

TL;DR

This paper develops a regularization method for Mickelsson generators in quantum groups associated with symplectic and orthogonal Lie algebras, ensuring these generators remain non-vanishing upon specialization at any weight.

Contribution

It introduces a regularization technique for Mickelsson generators in non-exceptional quantum groups, preventing their vanishing at any weight.

Findings

Regularized Mickelsson generators remain non-zero at all weights.

The method applies to quantum groups of symplectic and orthogonal types.

Ensures stable algebraic properties under specialization.

Abstract

Let $\mathfrak{g}'\subset \mathfrak{g}$ be the pair of Lie algebras of either symplectic or orthogonal infinitesimal endomorphisms of the complex vector spaces $\mathbb{C}^{N-2}\subset \mathbb{C}^N$ and $U_q(\mathfrak{g}')\subset U_q(\mathfrak{g})$ the pair of quantum groups with triangular decomposition $U_q(\mathfrak{g})=U_q(\mathfrak{g}_-)U_q(\mathfrak{g}_+)U_q(\mathfrak{h})$. Let $Z_q(\mathfrak{g},\mathfrak{g}')$ be the corresponding step algebra and regard its generators as rational trigonometric functions $\mathfrak{h}^*\to U_q(\mathfrak{g}_\pm)$. We describe their regularization such that the resulting generators do not vanish when specialized at any weight.