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On a class of \(n\)-Leibniz deformations of the simple Filippov algebras. (English) Zbl 1314.17004

Summary: We study the problem of 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 (FAs) 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 nontrivial one-parameter infinitesimal 3-Leibniz algebra deformation. We also show that the \(n \geqslant 3\) simple Filippov algebras do not admit nontrivial central extensions as \(n\)-Leibniz algebras of the above class.{
©2011 American Institute of Physics}

MSC:

17A40 Ternary compositions
17A42 Other \(n\)-ary compositions \((n \ge 3)\)
17A32 Leibniz algebras
17B37 Quantum groups (quantized enveloping algebras) and related deformations
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