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There are five real forms of the exceptional Lie group, $E_6$. Four of them are notated as in the following:

  • The split form as EI or $E_{6(6)}$

  • The quasi-split form as EII or $E_{6(2)}$

  • EIII or $E_{6(-14)}$

  • EIV or $E_{6(-26)}$

What do the annotations to $E_6$ actually indicate and are they also used for real forms for Lie groups in general?

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2 Answers 2

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The notation is a bit complicated to make precise, but the number in parentheses is the character, which is defined on page 353 section C of Helgason, Differential Geometry, Lie Groups and Symmetric Spaces. The character is the difference $\dim \mathfrak{p}_0 - \dim \mathfrak{k}_0$ in dimensions in a Cartan decomposition of a real form. As Helgason explains, the character of an exceptional real simple Lie algebra determines the Lie algebra, but for the classical real simple Lie algebras, it doesn't, so the notation is only used for exceptionals.

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$E_{6(n)}$ means that $n$ is the dimension of the group $E_6$ minus twice the dimension of a maximal compact subgroup.

example: $E_{6(6)}$ of dimension 78 has maximal compact subgroup ${\rm Sp}(4)/\mathbb{Z}_2$ of dimension 36.

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  • $\begingroup$ So could the universal cover of $E_{6(-14)}$ also be called $E_{6(-12)}$? $\endgroup$ Commented Aug 17, 2023 at 23:45
  • $\begingroup$ Most likely the definition is always $dim\mathfrak{p}-dim\mathfrak{k}$ to avoid that. $\endgroup$
    – Craig
    Commented Oct 19, 2023 at 9:43

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