We introduce the notion of a generalized metric n-Leibniz algebra and show that there is a one-to-one correspondence between generalized metric n-Leibniz algebras and faithful generalized orthogonal representations of metric Lie algebras (called Lie triple datas). | Turk J Math (2018) 42: 3061 – 3077 © TÜBİTAK doi: Turkish Journal of Mathematics Research Article Generalized metric n-Leibniz algebras and generalized orthogonal representation of metric Lie algebras Lina SONG∗,, Rong TANG , Department of Mathematics, Jilin University, Changchun, Jilin, . China Received: • Accepted/Published Online: • Final Version: Abstract: We introduce the notion of a generalized metric n -Leibniz algebra and show that there is a one-to-one correspondence between generalized metric n -Leibniz algebras and faithful generalized orthogonal representations of metric Lie algebras (called Lie triple datas). We further show that there is also a one-to-one correspondence between generalized orthogonal derivations (resp. generalized orthogonal automorphisms) on generalized metric n -Leibniz algebras and Lie triple data. Key words: Generalized metric n -Leibniz algebra, metric Lie algebra, generalized orthogonal representation, generalized orthogonal derivation, generalized orthogonal automorphism 1. Introduction Ternary Lie algebras (3-Lie algebras) or more generally n -ary Lie algebras are the natural generalization of Lie algebras. They were introduced and studied by Filippov in [13] and can be traced back to Nambu [22]. See [15–17, 23] and the review article [9] for more details. This type of algebras appeared also in the algebraic formulation of Nambu mechanics [22] and generalizing Hamiltonian mechanics by considering two Hamiltonians; see [14, 24]. Moreover, 3-Lie algebras appeared in string theory and M-theory. In [3], Basu and Harvey suggested replacimg the Lie algebra appearing in the Nahm equation by a 3-Lie algebra for the lifted Nahm equations. Furthermore, in the context of the Bagger–Lambert–Gustavsson model of multiple M2-branes, Bagger and Lambert managed to construct, using a ternary bracket, an N = 2 supersymmetric version of the world
