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*k*-Leibniz algebras from lower order ones: From Lie triple to Lie ℓ-ple systems

### Abstract

Two types of higher order Lie ℓ-ple systems are introduced in this paper. They are defined by brackets with ℓ > 3 arguments satisfying certain conditions, and generalize the well-known Lie triple systems. One of the generalizations uses a construction that allows us to associate a (2n − 3)-Leibniz algebra with a metric n-Leibniz algebra by using a 2(n − 1)-linear Kasymov trace form for . Some specific types of k-Leibniz algebras, relevant in the construction, are introduced as well. Both higher order Lie ℓ-ple generalizations reduce to the standard Lie triple systems for ℓ = 3.

© 2013 AIP Publishing LLC

Received 07 April 2013
Accepted 14 August 2013
Published online 17 September 2013

Acknowledgments:
The authors wish to thank Neil Lambert for a helpful conversation. This work has been partially supported by research grants from the Spanish MINECO (FIS2008-01980, FIS2009-09002, CONSOLIDER CPAN-CSD2007-00042).

Article outline:

I. INTRODUCTION
II. FILIPPOV AND *n*-LEIBNIZ ALGEBRAS
A. Filippov or *n*-Lie algebras
B. *n*-Leibniz algebras
C. Higher order Leibniz algebras of CS type
III. LIE TRIPLE SYSTEMS
IV. THE *k*-LEIBNIZ ALGEBRA ASSOCIATED WITH TWO *n*- AND *m*-LEIBNIZ ALGEBRAS
V. HIGHER ORDER LIE *k*-PLE SYSTEMS
A. Lie *n*-ple systems: A first generalization
B. Lie ℓ-ple systems, ℓ = 2*n* − 3
VI. CONCLUDING REMARKS