Integrable triples in semisimple Lie algebras
type: Item
instance of: scholarly article
title: Integrable triples in semisimple Lie algebras
author: Alberto De Sole
author: Victor G. Kac
author: Daniele Valeri
author: M. Jibladze
published in: Letters in Mathematical Physics
publication date: 2021-11-09T00:00:00Z
full work available at URL: https://arxiv.org/abs/2012.12913
review text: The present paper is a continuation of our paper [\textit{A. De Sole} et al., Transform. Groups 26, No. 2, 479--500 (2021; Zbl 07380376)], where we studied integrability of \(W\)-algebras. Take a nilpotent element \(f\) of a simple Lie algebra \(g\). By the Jacobson-Morozov theorem, any non-zero nilpotent \(f\) of a simple Lie algebra \(g\) can be included in an \(sl_2\)-triple \(s = \{e, f , h\}\). This produces a \(\mathbb{Z}\)-grading \(\oplus g_j\) of \(g\) by eigenspaces of \(adh\). In the paper cited above a notion of an integrable triple associated to the nilpotent element \(f\) was introduced. It is a triple of elements \(( f_1, f_2, E)\), where \(f_1, f_2\in g_{-2}, E\in g_j\) with \(j \geq 1\), such that \begin{itemize} \item[(I1)] \(f = f_1 + f_2\) and \([ f_1, f_2] = 0\); \item[(I2)] \([E, g_{\geq 2}] = 0\) and the centralizer of \(E\) in \(g_1\) is coisotropic with respect to the symplectic form \(\omega=(f|[a,b])\), where \((.|.)\) is a fixed non-degenerate symmetric invariant bilinear form on \(g\); \item[(I3)] \(f_1 + E\) is semisimple and \([ f_2, E] = 0\). \end{itemize} The first main result of the paper if Theorem 3.6, which states that a cyclic element \(f + E\) is not nilpotent if and only if the Zariski closure of the orbit \(Z(s)(E)\) does not contain \(0\). Here where \(Z(s)\) stands for the centralizer of the \(sl_2\)-triple \(s\) in the adjoint group \(G\) of \(g\). The second main result is Theorem 3.15, which states that a cyclic element \(f + E\) is integrable if and only if the orbit \(Z(s)(E)\) is closed.
Mathematics Subject Classification ID: 17B20
Mathematics Subject Classification ID: 17B25
Mathematics Subject Classification ID: 17B70
Mathematics Subject Classification ID: 17B80
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20G41
zbMATH DE Number: 7423095
zbMATH Keywords: integrable triple in a simple Lie algebra
zbMATH Keywords: cyclic and quasi-cyclic elements
zbMATH Keywords: polar representation
zbMATH Keywords: integrable hierarchy of Hamiltonian PDE
reviewed by: Dmitry Artamonov
describes a project that uses: SLA
MaRDI profile type: Publication
OpenAlex ID: W3199410170
Wikidata QID: Q114224883
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DOI: 10.1007/S11005-021-01456-4
description: scientific article; zbMATH DE number 7423095