On realizations of the complex Lie groups $ (E_{6,\mathbb{R}})^C, (E_{6,\mathbb{C}})^C, (E_{6,\mathbb{H}})^C $ and those real forms

TL;DR

This paper explores the realization of various real forms of the complex Lie group E6 by replacing algebraic structures with different fields such as real numbers, complex numbers, and quaternions, clarifying their structures.

Contribution

It systematically defines and analyzes the structure of E6 real forms through algebraic replacements, providing new insights into their realizations.

Findings

Defined new realizations of E6 real forms using different algebraic fields.

Clarified the structure of groups like (E_{6, ext{R}})^C and E_{6(-26), ext{H}}.

Extended understanding of Lie group realizations via algebraic substitutions.

Abstract

There exist six Lie groups of type $ E_6 $, and to be specific, ${E_6}^C , E_6, E_{6(6)}, E_{6(-2)}, E_{6(-14)}, E_{6(-26)}$. In order to define these groups, we use usually the Cayley algebra $ \mathfrak{C} $ and the split Cayley algebra $ \mathfrak{C}' $. In the present article, we consider the Lie groups which are defined by replacing $ \mathfrak{C}^C, \mathfrak{C} $ and $ \mathfrak{C}' $ with the fields of real numbers $\mathbb{R}$, complex numbers $\mathbb{C}$, split complex numbers $\mathbb{C}'$, quaternions $\mathbb{H}$ and split quaternions $\mathbb{H}'$. For instance, the group $(E_{6,\mathbb{R}})^C$ is given as a group defined by replacing $\mathfrak{C}$ with $\mathbb{R}$ in ${E_6}^C$ and the group $E_{6(-26),\mathbb{H}}$ is given as a group defined by replacing $\mathfrak{C}$ with $\mathbb{H}$ in $E_{6(-26)}$. We call { \it realization} to determine the structure of the group.