We introduce in this paper the contractions $\mathfrak{ G}_c$ of $n$-Lie (or Filippov) algebras $\mathfrak{ G}$ and show that they have a semidirect structure as their $n=2$ Lie algebra counterparts. As an example, we compute the non-trivial contractions of the simple $A_{ n+1}$ Filippov algebras. By using the \.In\"on\"u-Wigner and the generalized Weimar-Woods contractions of ordinary Lie algebras, we compare (in the $\mathfrak{ G}=A_{ n+1}$ simple case) the Lie algebras Lie$\,\mathfrak{ G}_c$ (the Lie algebra of inner endomorphisms of $\mathfrak{ G}_c$) with certain contractions $(\mathrm{ Lie}\,\mathfrak{ G})_{ IW}$ and $(\mathrm{ Lie}\,\mathfrak{ G})_{ W-W}$ of the Lie algebra Lie$\,\mathfrak{ G}$ associated with $\mathfrak{ G}$.
Categories: math-ph hep-th math.MP math.QA math.RA