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Another important application of algebraic methods in physics is to the study of exactly solvable problems in quantum mechanics. Consider quantum mechanics in v dimensions described by the Hamiltonian [ABSTRACT FROM AUTHOR]

One of the most important applications of Lie algebras has been to the study of physical systems for which the Hamiltonian H and other operators of physical interest T can be written in terms of elements Gα of a Lie algebra g [ABSTRACT FROM AUTHOR]

Lie algebras can be constructed with n x n square matrices [ABSTRACT FROM AUTHOR]

Lie algebras can also be constructed with differential operators acting on a space of derivable functions f(x) of the coordinate x. The basic commutation relations are [ABSTRACT FROM AUTHOR]

In Chap. 7 a realization of Lie algebras in terms of boson operators has been given. These operators have been used either in connection with coordinates and momenta, as in harmonic oscillator problems, see Chap. 7 [ABSTRACT FROM AUTHOR]

Realizations of Lie algebras in terms of boson operators are of great interest for applications to a variety of problems in physics, most notably to oscillator problems in quantum mechanics and to algebraic models of rotation-vibration spectra of molecules (vibron model) and nuclei (interaction boson model). [ABSTRACT FROM AUTHOR]

We introduce an irreducible basis β and write it generically as [ABSTRACT FROM AUTHOR]

An operator which commutes with all the elements of a Lie algebra, g, is called an invariant, or Casimir operator, C [ABSTRACT FROM AUTHOR]

An irreducible basis (IRB) is the basis for the representations of the algebra, g, (and the associated group, G), and the basis upon which the elements of the algebra [ABSTRACT FROM AUTHOR]

Associated with each Lie algebra there is a Lie group. The notion of groups of transformations precedes that of Lie algebras and is more general than it. A set of elements A,B, C., forms a group G if it satisfies the following axioms: [ABSTRACT FROM AUTHOR]

In 1895, the French mathematician Cartan provided a way to classify all semisimple Lie algebras. [ABSTRACT FROM AUTHOR]

The key notion in the de.nition of Lie algebras is that of the commutator (or bracket), denoted by [,]. [ABSTRACT FROM AUTHOR]