Journal of Algebra and Its Applications Vol. 11, No. 3 (2012) 1250062 (17 pages) c  World Scientific Publishing Company DOI: 10.1142/S0219498812500624 ON NILPOTENT LEIBNIZ n-ALGEBRAS L. M. CAMACHO ∗,¶ , J. M. CASAS †,∗∗ , J. R. G ´ OMEZ ∗,‖ , M. LADRA ‡,†† and B. A. OMIROV §,‡‡ ∗ Departamento de Matem´ atica Aplicada I, Universidad de Sevilla Avda. de Reina Mercedes s/n, 41012 Sevilla, Spain † Departamento de Matem´ atica Aplicada I, Universidad de Vigo EUIT. Forestal, Campus Universitario A Xunqueira 36005 Pontevedra, Spain ‡ Departamento de ´ Algebra, Universidad de Santiago E-15782 Santiago de Compostela, Spain § Institute of Mathematics and Information Technologies Uzbekistan Academy of Sciences, F. Hodjaev str. 29 100125 Tashkent, Uzbekistan ¶ lcamacho@us.es || jrgomez@us.es ∗∗ jmcasas@uvigo.es †† manuel.ladra@usc.es ‡‡ omirovb@mail.ru Received 24 December 2010 Accepted 29 August 2011 Communicated by I. Shestakov We study the nilpotency of Leibniz n-algebras related with the adapted version of Engel’s theorem to Leibniz n-algebras. We also deal with the characterization of finite- dimensional nilpotent complex Leibniz n-algebras. Keywords : Leibniz n-algebra; right nilpotency; right operator. Mathematics Subject Classification: 17A32 1. Introduction The simplest phase space for Hamiltonian mechanics is R 2 with coordinates x, y and canonical Poisson bracket {f 1 ,f 2 } = ∂f1 ∂x ∂f2 ∂y - ∂f1 ∂y ∂f2 ∂x = ∂(f1,f2) ∂(x,y) . This bracket satisfies the Jacobi identity {f 1 , {f 2 ,f 3 }} + {f 2 , {f 3 ,f 1 }} + {f 3 , {f 1 ,f 2 }} = 0 and gives rise to the Hamilton equations of motion. In 1973, Nambu [21] proposed the generalization of this example defining for a term of classical observables on the three-dimensional space R 3 with coordinates 1250062-1