2011/02/21 by J. A. de Azcarraga, J A de Azcárraga, J. M. izquierdo +1 · 4 citations
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebra over a field #Algebraic structures and combinatorial models #Homotopy and Cohomology in Algebraic Topology #Infinitesimal #Lemma (botany) #Poisson algebra #Poisson distribution #Simple (philosophy) #hep-th #math-ph #math.MP #math.QA #math.RA
paper · pdf · doi:10.1088/1742-6596/284/1/012019
published in Journal of Physics Conference Series 284, 012019 (IOP Publishing) · 11 pages
arxiv created 2011/02/21 · openalex publication_date 2011/03/01 · arxiv updated 2011/11/09 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
We describe the basic properties of two n -ary algebras, the Generalized Lie Algebras (GLAs) and, particularly, the Filippov (≡ n -Lie) algebras (FAs), and comment on their n -ary Poisson counterparts, the Generalized Poisson (GP) and Nambu-Poisson (N-P) structures. We describe the Filippov algebra cohomology relevant for the central extensions and infinitesimal deformations of FAs. It is seen that semisimple FAs do not admit central extensions and, moreover, that they are rigid. This extends the familiar Whitehead's lemma to all n ≥ 2 FAs, n = 2 being the standard Lie algebra case. When the n -bracket of the FAs is no longer required to be fully skewsymmetric one is lead to the n -Leibniz (or Loday's) algebra structure. Using that FAs are a particular case of n -Leibniz algebras, those with an anticommutative n -bracket, we study the class of n -Leibniz deformations of simple FAs that retain the skewsymmetry for the first n − 1 entires of the n -Leibniz bracket.