ABSTRACT
This chapter shows that some useful examples of finite-dimensional Lie algebras. There are many exactly solvable examples in quantum mechanics as well as in other fields of theoretical physics. A subset of a Lie algebra L that satisfies all the properties of a Lie algebra is a subalgebra of L. Finite-dimensional Lie algebras are far more tractable than those with infinite basis set. The exponential function of operators belonging to a Lie algebra appears in many physical applications, and it is commonly necessary to determine its matrix elements in a given basis set of state vectors. Typically, one accomplishes such a calculation and many others by rewriting a given exponential operator in a convenient way. For instance, one could be interested in expressing a single exponential as a product of properly chosen exponentials or vice versa. The parameter differentiation method may produce the remaining disentanglement relationships that the regular representation is unable to provide when it is unfaithful.
