Root spaces and Dynkin diagrams

Abstract

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. Completeness is obtained by introducing Dynkin diagrams. These specify the inner products among a fundamental set of basis roots in the root space diagram. In this approach completeness is relatively simple to prove, while enumeration of the remaining roots within a root space diagram is less so. Properties of roots In an effort to cast the commutation relations of a semisimple Lie algebra into an eigenvalue-eigenvector format, a secular equation was constructed from the regular representation. The rank of an algebra is, among other things: (i) the number of independent functions in the secular equation; (ii) the number of independent roots of the secular equation; (iii) the number of mutually commuting operators in the Lie algebra; (iv) the number of invariant operators that commute with all elements in the Lie algebra (Casimir operators); (v) the dimension of the positive-definite root space that summarizes the commutation relations.