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I ask as an undergraduate student the following questions, can I get some simple and satisfying answers ?

  1. what exactly is E8?

  2. how can I learn more about it ?

  3. in my little knowledge about lie groups I read that The Lie group SO(3) is diffeomorphic to the real projective space, is E8 a somewhat similar to that with the rotation matrix

$T=\begin{pmatrix} 0 &0 &0 &1 & & & & \\ 1 &0 &0 &0 & & & & \\ 0 &0 &1 &0 & & & & \\ 0 &1 &0 &0 & & & & \\ & & & & -\frac{1}{\sqrt{2}} &-\frac{\sqrt{3}}{2} & & \\ & & & & \frac{\sqrt{3}}{2} & -\frac{1}{2} & & \\ & & & & & &1 &0 \\ & & & & & & 0 &1 \end{pmatrix}$

“E8 is perhaps the most beautiful structure in all of mathematics, but it’s very complex.” — Hermann Nicolai

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    I think, the wikipedia article is worth reading, with many references. – Dietrich Burde Apr 30 '21 at 14:06
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    I guess one cannot really understand a specific Lie group without having more than "little knowledge" about Lie groups in general. – Torsten Schoeneberg Apr 30 '21 at 14:58
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    You started with one of the most complicated among compact (or complex) simple Lie group, which, I believe, is a wrong thing to do. My suggestion is first to learn simpler things, like representation theory of finite groups, then learn about Lie algebras and their relation to Lie groups, then root systems and classification of complex Lie algebras. Only then try E8. – Moishe Kohan May 01 '21 at 05:34
  • @MoisheKohan your comment on what to learn so I can eventually understand E8 was really helpful. If you have more suggestions or introductory books in mind about these things it would be a time saver for me. Nevertheless, thank you! – 領域展開 May 01 '21 at 11:06
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    There was a similar request here awhile ago. I second the recommendation of Hall's book in the accepted answer. The other suggestions are likely to be too advanced for you. – Moishe Kohan May 02 '21 at 18:03

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