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Lie Groups, Physics, and Geometry by Robert Gilmore

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10

Root spaces and Dynkin diagrams

 

 

In the previous chapter the canonical commutation relations for semisimple Lie algebras were elegantly expressed in terms of roots. Although roots were introduced to simplify the expression of commutation relations, they can be used to classify Lie algebras and to provide a complete list of simple Lie algebras. We achieve both aims in this chapter. However, we use two different methods to accomplish this. We classify Lie algebras by specifying their root space diagrams. This is a relatively simple job using a “building up” approach, adding roots to rank l root space diagrams to construct rank l + 1 root space diagrams. However, it is not easy to prove the completeness of root space diagrams by this method. ...

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