Contractions of Filippov algebras

Details Contractions of Filippov algebras Contractions of Filippov algebras

2010-09-02

arxiv.org/abs/1009.0372v3

J.Math.Phys.52:013516,2011

Physics

Mathematical Physics

plain latex, 36 pages. A few misprints corrected. This v3 is actually v2 (v1 had been replaced by itself by error). To appear

Scientific paper

10.1063/1.3533944

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 $\mathfrak{G}$.

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