Metric Lie n-algebras and double extensions

TL;DR

This paper establishes a comprehensive structure theorem for metric Lie n-algebras, showing they can be constructed from simple and one-dimensional cases through orthogonal sums and double extensions.

Contribution

It introduces the concept of double extension for metric Lie n-algebras and proves that all such algebras can be built from basic components using this operation and orthogonal sums.

Findings

All metric Lie n-algebras are obtained from simple and 1-dimensional algebras.

Double extension is a key operation for constructing metric Lie n-algebras.

The structure theorem generalizes known results for Lie algebras to n-algebras.

Abstract

We prove a structure theorem for Lie n-algebras possessing an invariant inner product. We define the notion of a double extension of a metric Lie n-algebra by another Lie n-algebra and prove that all metric Lie n-algebras are obtained from the simple and one-dimensional ones by iterating the operations of orthogonal direct sum and double extension.