Abstract
Publisher Summary This chapter provides information on quantum groups and their applications in the description of physical systems. It focuses on the quantum-algebraic analogue of the harmonic oscillator. In this approach, the analogues of the creation and annihilation operators of oscillator quanta, called “q-deformed boson creation” and “annihilation operators,” are introduced. This satisfies the commutation relations that slightly differ from the commutation relations for the standard boson creation and annihilation operators for the harmonic oscillator, but from a mathematical point of view this difference is very essential. Unlike the standard boson operators whose commutation relations close a Lie-algebra, the commutation relations of the q-deformed boson operators form Hopf algebra. However, the basic properties of the q-deformed boson operators can be deduced by using only their commutation relations and utilizing the fact that they form Hopf algebra only by implication.
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