Split 3-Lie algebras

Abstract

In order to begin an approach to the structure of 3-Lie algebras (with restrictions neither on the dimension nor on the base field), we introduce the class of split 3-Lie algebras as the natural extension of the class of split Lie algebras. By developing techniques of connections of roots for this kind of ternary algebras, we show that any of such split 3-Lie algebras \documentclass[12pt]{minimal}\begin{document}${\mathfrak T}$\end{document}T is of the form \documentclass[12pt]{minimal}\begin{document}${\mathfrak T}={\mathcal U} +\sum \limits _{j}I_{j}$\end{document}T=U+∑jIj with \documentclass[12pt]{minimal}\begin{document}${\mathcal U}$\end{document}U a subspace of the 0-root space \documentclass[12pt]{minimal}\begin{document}${\mathfrak T}_0$\end{document}T0 and any Ij a well described ideal of \documentclass[12pt]{minimal}\begin{document}${\mathfrak T}$\end{document}T, satisfying \documentclass[12pt]{minimal}\begin{document}$[I_j,{\mathfrak T},I_k]=0$\end{document}[Ij,T,Ik]=0 if j ≠ k. Under certain conditions the simplicity of \documentclass[12pt]{minimal}\begin{document}${\mathfrak T}$\end{document}T is characterized and it is shown that \documentclass[12pt]{minimal}\begin{document}${\mathfrak T}$\end{document}T is the direct sum of the family of its minimal ideals, each one being a simple split 3-Lie algebra.