Year: 2022
Type: preprint
Abstract: Shapovalov elements $θ_{β,m}$ are special elements in a Borel subalgebra of a classical or quantum universal enveloping algebra parameterized by a positive root $β$ and a positive integer $m$. They relate the canonical generator of a reducible Verma module with highest vectors of its Verma submodules. For $m=1$, they can be explicitly obtained as matrix elements of the inverse Shapovalov form. We extend this approach to $m>1$ for all $β$ but three roots in $\mathfrak{g}_2$, $\mathfrak{f}_4$, and $\mathfrak{e}_8$, presenting $θ_{β,m}$ as a product of matrix elements of weight $β$.
Source: arXiv (Cornell University)
Author: Mudrov, Andrey
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4226436336