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 ${\mathfrak{C}}^{\prime }$. In the present article, we consider the Lie groups which are defined by replacing ${\mathfrak{C}}^C,\mathfrak{C}$ and ${\mathfrak{C}}^{\prime }$ with the fields of real numbers $\mathbb{R}$, complex numbers $\mathbb{C}$, split complex numbers ${\mathbb{C}}^{\prime }$, quaternions $\mathbb{H}$ and split quaternions ${\mathbb{H}}^{\prime }$. 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.

36 pages. arXiv admin note: text overlap with arXiv:2409.07760