On a class of n-Leibniz deformations of the simple Filippov algebras

Details On a class of n-Leibniz deformations of the simple Filippov algebras On a class of n-Leibniz deformations of the simple Filippov algebras

2010-09-14

arxiv.org/abs/1009.2709v2

J.Math.Phys.52:023521,2011

Physics

Mathematical Physics

19 pages, 30 refs., no figures. Some text rearrangements for better clarity, misprints corrected. To appear in J. Math. Phys

Scientific paper

10.1063/1.3553797

We study the problem of the infinitesimal deformations of all real, simple, finite-dimensional Filippov (or n-Lie) algebras, considered as a class of n-Leibniz algebras characterized by having an n-bracket skewsymmetric in its n-1 first arguments. We prove that all n>3 simple finite-dimensional Filippov algebras are rigid as n-Leibniz algebras of this class. This rigidity also holds for the Leibniz deformations of the semisimple n=2 Filippov (i.e., Lie) algebras. The n=3 simple FAs, however, admit a non-trivial one-parameter infinitesimal 3-Leibniz algebra deformation. We also show that the $n\geq 3$ simple Filippov algebras do not admit non-trivial central extensions as n-Leibniz algebras of the above class.

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