On the derived algebra of a centraliser

2010/03/02 by Oksana Yakimova, Yakimova, Oksana
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #math.RT

paper · pdf · doi:10.48550/arxiv.1003.0602

arxiv created 2010/03/02 · arxiv updated 2010/03/05

Abstract

Let \g be a classical Lie algebra, e a nilpotent of \g element and \gt g_e the centraliser of e in \g. We prove that \g_e=[\g_e,\g_e] if and only if e is rigid. It is also shown that if e is contained in [\g_e,\g_e], then the nilpotent radical of \g_e coincides with [\g(1)_e,\g_e], where \g(1)_e is an eigenspace of a characteristic of e with the eigenvalue 1.

Related

• Surprising properties of centralisers in classical Lie algebras
• The centralisers of nilpotent elements in classical Lie algebras
• Levi decomposition of nilpotent centralisers in classical groups
• The index of a Lie algebra, the centraliser of a nilpotent element, and the normaliser of the centraliser
• On symmetric invariants of centralisers in reductive Lie algebras