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If anyone has a copy of this book, I'm just struggling to understand a point in the chapter about Lie Groups. In particular the section on qunatization of the roots, on p177-178, he generates the diagrams in fig 9.2 from those of 9.1. I thought I understand this, but I don't understand how he gets from (c) in 9.1 to (c) in 9.2, I mean we have roots strings of length 2 and 3 for [tex]\alpha[/tex] strings through [tex] \beta [/tex] and [tex]\beta[/tex] strings through [tex] \alpha [/tex], then the two original roots themselves. So why don't we have 2+2+3=7 roots in total? the diagram (c) in fig 9.2 appears to show 8 in total?

If we look at going from (b) 9.1 to 9.2, this logic seems to work, we start with 2, each root string is length two, giving a total of 2+2+2=6 roots, sim for (a) going from 9.1 to 9.2, 2 originally, each root string of length 1, giving 2+1+1=4 roots in total.

Not sure if I'm understanding what he is doing correctly, I don't really understand why he is allowed to set p=0 either, or equivalently assume [tex] [E_{\beta},E_{\alpha}]=0[/tex]

Thanks for any help

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# Roots of Lie Algebra (Jones book)

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