PreprintCC BY 4.0arXiv
Geometry and coefficients of Šapovalov elements for KM Lie superalgebras
Abstract
We study Shapovalov elements for a symmetrizable, integrable Kac-Moody Lie superalgebra g. Let γ be a positive root of 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 θγ, m vλ 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.
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BibTeX
@misc{musson2026geometry,
title = {{Geometry and coefficients of Šapovalov elements for KM Lie superalgebras}},
author = {Ian M. Musson},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.29155},
eprint = {2607.29155},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.29155},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Geometry and coefficients of Šapovalov elements for KM Lie superalgebras
AU - Ian M. Musson
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2607.29155
UR - https://arxiv.org/abs/2607.29155
AB - We study Shapovalov elements for a symmetrizable, integrable Kac-Moody Lie superalgebra g. Let γ be a positive root of 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.
ER -