Shapovalov Elements For $U_q(\mathfrak{sl}(N+1))

TL;DR

This paper extends explicit formulas for Shapovalov elements from classical to quantum groups, specifically for the algebra alg(sl(N+1)), enhancing understanding of highest weight vectors in quantum Verma modules.

Contribution

It adapts the classical construction of Shapovalov elements to the quantum setting for alg(sl(N+1)), providing explicit expressions in the quantized enveloping algebra.

Findings

Explicit formulas for Shapovalov elements in the quantum case

Extension of classical results to quantum groups

Improved tools for studying highest weight modules

Abstract

For a simple Lie algebra, Shapovalov elements give rise to highest weight vectors in Verma modules. The usual construction of these elements uses induction on the length of a certain Weyl group element. If $\mathfrak{g}= \mathfrak{sl}(N+1)$ explicit expressions for Shapovalov elements were given in [Mus22a]. Here we adapt the argument to the quantized enveloping algebra of $\mathfrak{g}$.