Generalized metric $n$-Leibniz algebras and generalized orthogonal representation of metric Lie algebras

TL;DR

This paper introduces generalized metric n-Leibniz algebras and establishes their correspondence with faithful generalized orthogonal representations of metric Lie algebras, enriching the algebraic structure theory.

Contribution

It defines generalized metric n-Leibniz algebras and links them to orthogonal representations of metric Lie algebras, providing a new framework for understanding their structure.

Findings

One-to-one correspondence between generalized metric n-Leibniz algebras and Lie triple data.

Correspondence between orthogonal derivations/automorphisms and Lie triple data.

Framework for classifying generalized metric n-Leibniz algebras.

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 data). 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 datas.