ar X iv : m at h / 99 02 14 1 v 1 [ m at h . Q A ] 2 4 Fe b 19 99 Braided Oscillators

A generalized oscillator algebra is proposed and the braided Hopf algebra structure for this generalized oscillator is investigated. Using the solutions for the braided Hopf algebra structure, two types of braided Fibonacci oscillators are introduced. This leads to two types of braided Biedenharn-Macfarlane oscillators as special cases of the Fibonacci oscillators. We also find the braided Hopf algebra solutions for the three dimensional braided space. One of these, as a special case, gives the Hopf algebra given in the literature. The harmonic oscillator has a wide variety of applications from quantum optics to the realizations of the angular momentum algebra and hence the deformations of the oscillator algebra play an important role in q-deformed theories. The realization of the q-deformed angular momentum algebra by Biedenharn-Macfarlane oscillators [1] and the realization of the Hermitian braided matrices by a pair of q-oscillators [2] are some of the examples. The two parameter deformations and some of their applications can be found in [3]. Braided group theory (a self contained review can be found in [4]) deforms the notion of tensor product (called braided tensor product) and hence deforms the independence of the objects. Although braided groups arise in the formulation of quantum group covariant structures, the idea of braiding can be used without any reference to quantum groups to generalize the statistics [5]. The permutation map π (π : A ⊗ B → B ⊗ A) in the tensor product algebra of boson algebras (a ⊗ b)(c ⊗ d) = aπ(b ⊗ c)d = ac ⊗ bd is replaced by a generalized map called braiding ψ (ψ : A ⊗ B → B ⊗ A) such that (a ⊗ b)(c ⊗ d) = aψ(b ⊗ c)d.