Contractions of Filippov algebras

TL;DR

This paper explores the contraction processes of n-Lie (Filippov) algebras, establishing their semidirect structure, and compares these contractions with classical Lie algebra contractions, providing explicit examples with simple algebra cases.

Contribution

It introduces contraction methods for n-Lie algebras, analyzes their structure, and compares these with traditional Lie algebra contractions, especially for simple algebra cases.

Findings

Contractions of n-Lie algebras have a semidirect structure similar to Lie algebras.

Explicit computation of non-trivial contractions for simple $A_{n+1}$ Filippov algebras.

Comparison between algebraic contractions of Filippov algebras and classical Lie algebra contractions.

Abstract

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 $\mathfrak{G}$.