Contractions of Filippov algebras

Abstract

We introduce in this paper the contractions \documentclass[12pt]{minimal}\begin{document}$\mathfrak {G}_c$\end{document}Gc of n-Lie (or Filippov) algebras \documentclass[12pt]{minimal}\begin{document}$\mathfrak {G}$\end{document}G and show that they have a semidirect structure as their n = 2 Lie algebra counterparts. As an example, we compute the nontrivial contractions of the simple \documentclass[12pt]{minimal}\begin{document}$A_{n+1}$\end{document}An+1 Filippov algebras. By using the İnönü–Wigner and the generalized Weimar-Woods contractions of ordinary Lie algebras, we compare (in the \documentclass[12pt]{minimal}\begin{document}$\mathfrak {G}=A_{n+1}$\end{document}G=An+1 simple case) the Lie algebras Lie\documentclass[12pt]{minimal}\begin{document}$\,\mathfrak {G}_c$\end{document}Gc (the Lie algebra of inner endomorphisms of \documentclass[12pt]{minimal}\begin{document}$\mathfrak {G}_c$\end{document}Gc) with certain contractions \documentclass[12pt]{minimal}\begin{document}$({\rm Lie}\,\mathfrak {G})_{IW}$\end{document}( Lie G)IW and \documentclass[12pt]{minimal}\begin{document}$({\rm Lie}\,\mathfrak {G})_{W-W}$\end{document}( Lie G)W−W of the Lie algebra Lie\documentclass[12pt]{minimal}\begin{document}$\,\mathfrak {G}$\end{document}G associated with \documentclass[12pt]{minimal}\begin{document}$\mathfrak {G}$\end{document}G.