I have some questions regarding the exceptional Lie algebras e(n), n=6,7,8.(adsbygoogle = window.adsbygoogle || []).push({});

Can anybody explain to me what prevents us from constructing e(9) from e(8)? What goes wrong? One can use the e(8) lattice vectors and try to construct an e(9) vector; one could go even further and try e(10) etc. I know that the Cartan Matrix becomes zero (or negative for 10, ...) in that case (which is forbidden), but what does that mean if one would try to write down the generators for e(9)? What's wrong with them as Lie algebra generators?

Another question I have is related to E(n) as symmetry groups. For the A, B, C and D series one can understand the (fundamental or defining representation of) Lie groups acting on a certain vector space and leaving a certain scalar product invariant. For SO(n) it's

[tex]\vec{x}^t \vec{y}[/tex]

with [tex]\vec{x}, \vec{y} \in R^n[/tex], for SU(n) it's

[tex](\vec{x}^\ast)^t \vec{y}[/tex]

with [tex]\vec{x}, \vec{y} \in C^n[/tex]. What about E(n)? Is there a similar scalar product which is invariant? Are there other invariants?

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# Exceptional Lie algebras

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