I have some questions regarding the exceptional Lie algebras e(n), n=6,7,8.
Can anybody explain to me what prevents us from constructing e(9) from e(8)? 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, …) 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? Where does the re-construction of the Lie algebra from the Dynkin diagram / Cartan matrix fail?
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 (x,y) with x,y living in a real vector space, for SU(n) it's (x*, y) with x,y living in a complex vector space. What about E(n)? Is there a similar scalar product which is invariant? What is the corresponding vector space? Are there other invariants?
(my background is theoretical physics, particle physics, gauge symmetries etc.)
Thanks
Tom
Best Answer
See http://en.wikipedia.org/wiki/En_%28Lie_algebra%29
Nothing goes wrong when you construct a Lie algebra E10, E11, ... by generators and relations from the Cartan matrix. The only difference is that the Lie algebras you get are infinite dimensional. E9 is a central extension of the affine E8 algebra, but E10 and beyond seem rather a mess.
E6, E7, E8 can be represented as symmetry groups of various forms on spaces of dimensions 27, 56, and 248. E9 and beyond act naturally on infinite dimensional vertex algebras, but apart from E9 are not the full symmetry groups of these algebras.