Integrable triples in semisimple Lie algebras
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.
integrable triple in a simple Lie algebra
scientific article; zbMATH DE number 7423095