Root vector localization and minimal nilpotent finite W-algebras
Abstract
Let g be any finite-dimensional complex simple Lie algebra, let h∨ be its dual Coxeter number, and let e be its highest-root vector which is a minimal nilpotent element. In this paper, using a result of Bavula on invariants of commuting locally nilpotent derivations, we show that U(g)f ≅ 𝒟+h∨−2⊗ 𝒟1⊗ U(g,e), where the first factor is a polynomial Weyl algebra of rank h∨−2, the second factor is a Laurent Weyl algebra of rank one, and U(g,e) is the finite W-algebra associated with g and e. As an application, we prove that if M is a simple infinite-dimensional g-module with the minimal GK dimension on which f−1 acts locally nilpotently, then M≅ P⊗ V for some simple 𝒟+h∨−2⊗ 𝒟1-module P and some finite-dimensional simple U(g,e)-module V. We also give a new proof of the Skryabine quivalence and Petukhov's birational decomposition of the quotient division ring of U(g).
Related
- A non-nilpotent localization of a finite p-group
- Finite group actions on the non-minimal nilpotent orbit closures in type A
- Local Derivations and Local Automorphisms of Arbitrary Finite-Dimensional Nilpotent Lie and Leibniz Algebras
- Minimal presentations of finite-dimensional representations of pseudocompact algebras
- Even and odd minimal categorifications of the nilpotent part of classical and quantum sl(2)
BibTeX
@misc{liu2026root,
title = {{Root vector localization and minimal nilpotent finite $W$-algebras}},
author = {Genqiang Liu},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05855},
eprint = {2611.05855},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05855},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Root vector localization and minimal nilpotent finite W-algebras
AU - Genqiang Liu
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.05855
UR - https://arxiv.org/abs/2611.05855
AB - Let g be any finite-dimensional complex simple Lie algebra, let h^∨ be its dual Coxeter number, and let e be its highest-root vector which is a minimal nilpotent element. In this paper, using a result of Bavula on invariants of commuting locally nilpotent derivations, we show that U(g)_f ≅ 𝒟^+_h^∨−2⊗ 𝒟_1⊗ U(g,e), where the first factor is a polynomial Weyl algebra of rank h^∨−2, the second factor is a Laurent Weyl algebra of rank one, and U(g,e) is the finite W-algebra associated with g and e. As an application, we prove that if M is a simple infinite-dimensional g-module with the minimal GK dimension on which f−1 acts locally nilpotently, then M≅ P⊗ V for some simple 𝒟^+_h^∨−2⊗ 𝒟_1-module P and some finite-dimensional simple U(g,e)-module V. We also give a new proof of the Skryabine quivalence and Petukhov's birational decomposition of the quotient division ring of U(g).
ER -