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

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.