Year: 2026
Type: preprint
Abstract: We study Shapovalov elements for a symmetrizable, integrable Kac-Moody Lie superalgebra $\mathfrak{g}$. Let $γ$ be a positive root of $\mathfrak{g}$ and $m$ a positive integer, with suitable conditions on the pair $(γ, m).$ The Shapovalov element $θ_{γ,m}\in U(\mathfrak{b}^-) $ has the important property that if $λ$ lies on a certain hyperplane, then $θ_{γ, m} v_λ$ is either zero or a highest weight vector of weight $λ-mγ$ in $M(λ)$. We give a closed fomula for all coefficients that arise in the induction step of the proof. The commutative version of this formula relates the leading terms using maps graded algebras that we call of Shapovalov algebras. This suggests a geometric setting for the study of Shapovalov elements.
Source: arXiv (Cornell University)
Author: Musson, Ian M.
Institution: University of Wisconsin–Milwaukee
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7172264414