On a quarternification of complex Lie algebras

TL;DR

This paper introduces the concept of quarternification of complex Lie algebras, defining quaternion Lie algebras and proving that simple Lie algebras admit such quarternifications, with detailed root space decompositions.

Contribution

It defines quaternion Lie algebras and establishes that every simple Lie algebra has a quarternification, extending classical Lie algebra theory.

Findings

gl(n,H), sl(n,H), so*(2n), sp(n) are quarternifications of classical complex Lie algebras.

Root space decomposition of quarternified Lie algebras is provided.

Each fundamental root space is complex 2-dimensional.

Abstract

We give a definition of quarternion Lie algebra and of the quarternification of a complex Lie algebra. By our definition gl(n,H), sl(n,H), so*(2n) and sp(n) are quarternifications of gl(n,C), sl(n,C), so(n,C) and u(n) respectively. Then we shall prove that a simple Lie algebra admits the quarternification. For the proof we follow the well known argument due to Harich-Chandra, Chevalley and Serre to construct the simple Lie algebra from its corresponding root system. The root space decomposition of this quarternion Lie algebra will be given. Each root space of a fundamental root is complex 2-dimensional.