Abstract. Deformed orthogonal and pseudo-orthogonal Lie algebras are constructed which differ from deformations of Lie algebras in terms of Cartan subalgebras and root vectors and which make it possible to construct representations by operators acting according to Gel'fand-Tsetlin-type formulas. Unitary representations of the q-deformed algebras U q (so n,1 ) are found.

Composite bosons, here called quasibosons (e.g. mesons, excitons, etc.) occur in various physical situations. Quasibosons differ from bosons or fermions as their creation and annihilation operators obey non-standard commutation relations, even for the "fermion+fermion" composites. Our aim is to realize the operator algebra of quasibosons composed of two fermions or two q-fermions (q-deformed fermions) by the respective operators of deformed oscillators, the widely studied objects. For this, the restrictions on quasiboson creation/annihilation operators and on the deformed oscillator (deformed boson) algebra are obtained. Their resolving proves the uniqueness of the family of deformations and gives explicitly the deformation structure function (DSF) which provides the desired realization. In the case of two fermions as constituents, such realization is achieved when the DSF is quadratic polynomial in the number operator. In the case of two q-fermions, q = 1, the obtained DSF inherits the parameter q and does not continuously converge when q → 1 to the DSF of the first case.

We study main features of the exotic case of q-deformed oscillators (so-called TammDancoff cutoff oscillator) and find some special properties: (i) degeneracy of the energy levels En 1 = E n 1 +1 , n 1 ≥ 1, at the real value q = q n 1 n 1 +2of deformation parameter, as well as the occurrence of other degeneracies En 1 = E n 1 +k , for k ≥ 2, at the corresponding values of q which depend on both n 1 and k; (ii) the position and momentum operators X and P commute on the state |n 1 if q is fixed as q = n 1 n 1 +1, that implies unusual uncertainty relation; (iii) two commuting copies of the creation, annihilation, and number operators of this q-oscillator generate the corresponding q-deformation of the non-simple Lie algebra su(2) ⊕ u(1) whose nontrivial q-deformed commutation relation is: [J + , J − ] = 2J 0 q 2J 3 −1 where J 0 ≡ 1 2 (N 1 − N 2 ) and J 3 ≡ 1 2 (N 1 + N 2 ).

Abstract. q-deformed oscillators and the q-Bose gas model enable effective description of the observed non-Bose type behavior of the intercept ("strength") λ (2) ≡ C (2) (K, K) − 1 of two-particle correlation function C (2) (p 1 , p 2 ) of identical pions produced in heavy-ion collisions. Three-and n-particle correlation functions of pions (or kaons) encode more information on the nature of the emitting sources in such experiments. And so, the q-Bose gas model was further developed: the intercepts of n-th order correlators of q-bosons and the n-particle correlation intercepts within the q,p-Bose gas model have been obtained, the result useful for quantum optics, too. Here we present the combined analysis of two-and three-pion correlation intercepts for the q-Bose gas model and its q,p-extension, and confront with empirical data (from CERN SPS and STAR/RHIC) on pion correlations. Similar to explicit dependence of λ (2) on mean momenta of particles (pions, kaons) found earlier, here we explore the peculiar behavior, versus mean momentum, of the 3-particle correlation intercept λ (3) (K). The whole approach implies complete chaoticity of sources, unlike other joint descriptions of two-and three-pion correlations using two phenomenological parameters (e.g., core-halo fraction plus partial coherence of sources).

Abstract. The approach based on multimode system of q-deformed oscillators and the related picture of ideal gas of q-bosons enables to effectively describe the observed non-Bose type behaviour, in experiments on heavy-ion collisions, of the intercept (or the "strength") λ of the two-particle correlation function of identical pions or kaons. In this paper we extend main results of that approach in the two aspects: first, we derive in explicit form the intercepts of n-particle correlation functions in the case of q-Bose gas model and, second, provide their explicit two-parameter (or qp-) generalization.