The aim of this paper is to show that there is a Hopf structure of the parabosonic and parafermionic algebras and this Hopf structure can generate the well known Hopf algebraic structure of the Lie algebras, through a realization of Lie algebras using the parabosonic (and parafermionic) extension of the Jordan Schwinger map. The differences between the Hopf algebraic and the graded Hopf superalgebraic structure on the parabosonic algebra are discussed. . The realization of a Lie algebra by bosons corresponds to the symmetric representation of the Lie algebra, while the realization by using fermions corresponds to the antisymmetric representation of the Lie algebra. The case of the u(N) algebra was presented in [3]. The bosons and the fermions are special cases of parabosons and parafermions, which are introduced by Green [4]. The parabosonic (and correspondingly the parafermionic) algebra is a generalization of the usual bosonic (fermionic) algebra leading to generalized alternatives of the Bose -Einstein (FermiDirac) statistics or to field theories based on paraparticles, all the related bibliography and details can be found in [5]. The N parabosons were used for constructing realization of a sp(2N) algebra , this construction is based on the idea of using parabosons, rather than usual bosons. The same idea can be applied in the case of the so(2N) algebra but N parafermions are used [5][ §3.2]. Biswas and Soni [6] used systematically parabosons or parafermions for a Jordan -Schwinger realization of a u(N) algebra. In the same paper realizations of the so(2N + 1) or sp(2N) algebras, using parafermions and parabosons, were discussed in a similar way as in [5,7,8]. Also a realization of the g(M/N) super algebra is proposed by using M parafermionic and N parabosonic operators, by extending the corresponding realizations based on the use of usual fermions and bosons. Palev [9] has shown that, the bilinear combinations of the paraoperators yield the superalgebra gl(n/m) (see also in the same paper the realizations of so(2n+1) and of osp(1/2m)). Later the same author [10] has also proved that the parabosonic and parafermionic algebras can be used for constructing realizations of osp(2N + 1/2M) algebras.The extension of the Jordan -Schwinger map as a method of a realization of every Lie algebra, using parabosons and parafermions was originally published since 1971 in a local journal by Palev[11]. This work is not widely known, even we ignored it, when the first version of this work was printed as a preprint.The fact that Lie algebras and superalgebras have a Hopf algebra structure, constitutes a strong indication that the parabosonic and parafermionic algebras might possess a Hopf algebraic structure too. If this is true, then the Hopf algebra structure of the Lie algebra should be consistent with the supposed Hopf algebra structure of the parabosonic and parafermionic al-

We consider a scalar thick brane configuration arising in a 5D theory of gravity coupled to a self-interacting scalar field in a Riemannian manifold. We start from known classical solutions of the corresponding field equations and elaborate on the physics of the transverse traceless modes of linear fluctuations of the classical background, which obey a Schrödinger-like equation. We further consider two special cases in which this equation can be solved analytically for any massive mode with m 2 ≥ 0, in contrast with numerical approaches, allowing us to study in closed form the massive spectrum of Kaluza-Klein (KK) excitations and to analytically compute the corrections to Newton's law in the thin brane limit. In the first case we consider a novel solution with a mass gap in the spectrum of KK fluctuations with two bound states -the massless 4D graviton free of tachyonic instabilities and a massive KK excitation -as well as a tower of continuous massive KK modes which obey a Legendre equation. The mass gap is defined by the inverse of the brane thickness, allowing us to get rid of the potentially dangerous multiplicity of arbitrarily light KK modes. It is shown that due to this lucky circumstance, the solution of the mass hierarchy problem is much simpler and transparent than in the thin Randall-Sundrum (RS) two-brane configuration. In the second case we present a smooth version of the RS model with a single massless bound state, which accounts for the 4D graviton, and a sector of continuous fluctuation modes with no mass gap, which obey a confluent Heun equation in the Ince limit. (The latter seems to have physical applications for the first time within braneworld models). For this solution the mass hierarchy problem is solved with positive branes as in the Lykken-Randall (LR) model and the model is completely free of naked singularities. We also show that the scalar-tensor system is stable under scalar perturbations with no scalar modes localized on the braneworld configuration.

Bosons and parabosons are described as associative superalgebras, with an infinite number of odd generators. Bosons are shown to be a quotient superalgebra of parabosons, establishing thus an even algebra epimorphism which is an immediate link between their simple modules. Parabosons are a super-Hopf algebra. The super-Hopf algebraic structure of parabosons, combined with the projection epimorphism previously stated, provides a braided interpretation of the Green ansatz device and of the parabosonic Fock-like representations. This braided interpretation combined with an old problem leads to the construction of straightforward generalizations of the Green ansatz.

We will present and study an algebra describing a mixed paraparticle model, known in the bibliography as "The Relative Parabose Set (Rpbs)". Focusing in the special case of a single parabosonic and a single parafermionic degree of freedom P(1,1) BF , we will construct a class of Fock-like representations of this algebra, dependent on a positive parameter p a kind of generalized parastatistics order. Mathematical properties of the Fock-like modules will be investigated for all values of p and constructions such as ladder operators, irreducibility (for the carrier spaces) and (Z2 × Z2)-gradings (for both the carrier spaces and the algebra itself) will be established.