I've just started learning about Lie theory (only just finished up to basic classification of semisimple lie algebras) and I've got the following questions:

How do I show that the complex lie algebra of type $E_6$ has 78 dimensions?

What is the fundamental representation of $E_6$ group and why does it has 27 dimensions?

I understand that these questions might seems a bit annoying to answer so if there's no simple answer could someone recommend me any readings for further lie theory that will focus on exceptional lie algebras/groups? (preferably a maths textbook/lecture note but with some application to physics as well because that's ultimately what I will be doing).

Thank you!


The answer depends on some facts about the classifcation of complex simple Lie algebras. We can use the formula $$ \dim (L)=\mid\Phi\mid+rank(L) $$ for a simple Lie algebra $L$, where $\mid\Phi\mid$ denotes the cardinality of the root system of $L$. Now we can list explicitly $72$ roots of $L=E_6$, see here, with a picture of all $72$ roots. Because obviously $rank(E_6)=6$ this gives $$ \dim(E_6)=72+6=78. $$ As for the fundamental representation, Weyl's dimension formula gives $1$ and $27$ as the smallest dimensions, and it is easy to construct these representations; in particualr if we realize $E_6$ as a derivation algebra using a $27$-dimensional exceptional Jordan algebra.


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