General Discussion of Lie Algebras

It should be emphasized that a detailed knowledge of Lie algebras is not essential to the understanding of the applications we shall consider, since all of our results will be presented in a simple pedagogical manner using only the familiar concepts of vector spaces, operators, matrices, and commutators (see, e.g., Hamermesh, 1962 Saletan and Cromer, 1971). 

A Lie algebra over a field F is a vector space V over F equipped with a law of composition (Lie multiplication) denoted by [A, B] which has the following properties  

The first property (1) guarantees the closure of the Lie algebra under the Lie multiplication. Properties (2) and (3) combine this multiplication with the multiplication by the scalars of the vector space V, and imply the usual bilinearity properties 

In quantum mechanics we often encounter associative algebras of operators and matrices which are noncommutative. For example, the set of all n x n matrices over the real or complex number fields is an n2-dimensional vector space which is also an associative, noncommutative algebra whose multiplication is just the usual matrix multiplication. Also, the subset of all diagonal n x n matrices is a commutative algebra. 

Lie algebras can often be constructed from associative algebras of operators or matrices. In fact, the Lie algebras we shall consider for physical applications can all be constructed in this manner. Thus, given an associative algebra with multiplication defined by AB we can define the Lie product by the commutator, or Lie bracket of A and B 

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