Between Snyder's quantized space-time model in de Sitter space of momenta and the dS special relativity on dS-spacetime of radius R with Beltrami coordinates, there is a one-to-one dual correspondence supported by a minimum uncertainty-like argument. Together with Planck length ℓ P , R ≃ (3/Λ) 1/2 should be a fundamental constant. They lead to a dimensionless constant g∼ ℓ P R −1 = (G c −3 Λ/3) 1/2 ∼ 10 −61 . These indicate that physics at these two scales should be dual to each other and there is in-between gravity of local dS-invariance characterized by g. A simple model of dS-gravity with a gauge-like action on umbilical manifolds may show these characters. It can pass the observation tests and support the duality.

Based on the principle of relativity and the postulate on universal invariant constants (c, l), all possible kinematics can be set up with sub-symmetries of the UmovWeyl-Fock transformations for the inertial motions. Further, in the combinatory approach, all these symmetries are intrinsically related to each other, e.g. to the very important dS kinematics for the cosmic scale physics.

Based on the principle of relativity and the postulate on universal invariant constants (c, l) as well as Einstein's isotropy conditions, three kinds of special relativity form a triple with a common Lorentz group as isotropy group under full Umov-WeylFock-Lorentz transformations among inertial motions.

We show that if Yang's quantized space-time model is completed at both classical and quantum level, it should contain both Snyder's model, the de Sitter special relativity and their duality. PACS numbers: 03.30.+p, 02.40.Dr, 04.60.-m, 04.90.+e Contents I. Introduction 1 II. A Complete Yang Model and a UV-IR Duality 3 III. The Snyder's Model-dS special relativity Duality from the Yang Model 4 A. Snyder's model from the Yang model 4 B. The Beltrami model of dS special relativity from the Yang model 6 C. The Snyder's model-dS special relativity duality in the Yang model 8 IV. Concluding Remarks 9 Acknowledgments 9 References 9 * Electronic address: hyguo@itp.ac.cn † Electronic address: huangcg@mail.ihep.ac.cn ‡ Electronic address: lobby_wu@yahoo.com.cna, a scale near or at the Planck length ℓ P . The 4-energy-momentum were defined by the inhomogeneous projective coordinates. Then, Snyder identified the space-time coordinates' noncommutative operatorsx µ with 4-'translation' generators of a dS-algebra so(1, 4) and other operators as angular momentum for an so(1, 3) ⊂ so(1, 4).Soon after, Yang [2] extended Snyder's model to the one with the third constant, the radius R of a dS universe in order to recover the translation under R → ∞. Yang found an so(1, 5) algebra with c, a and R in a 6-d space with Minkowski (Mink) signature. In Yang's algebra, there are two so(1, 4) subalgebras for coordinate operatorsx µ and momentum operatorsp µ , respectively, with a common so(1, 3) for angular momentum operatorsl µν . And the algebra is invariant under a Z 2 dual transformation between a,x µ and /R,p µ . This is a UV-IR parameters' transformation.Recently, the 'doubly special relativity' or the 'deformed special relativity' (DSR) has been proposed [3]. There is also a universal constant κ near the Planck energy, related to /a in Snyder's model in addition to c. In the sense with one more universal constant a near or at the Planck length ℓ P in addition to c, Snyder's model might be regarded as the earliest DSR and some of DSR models can be given as the generalization of Snyder's model [4]. Soon, the 'triply special relativity' (TSR) [5] has also been proposed under a deformed symmetry with one more universal constant, the universe radius R ∼ Λ −1/2 . Later, it is found that the Lie algebra form of the deformed symmetry is just Yang's algebra in the 6-d space [6]. Thus, Yang's model might be regarded as a TSR (see also [7,8]).The projective geometry approach to dS-space is basically equivalent to the Beltrami model of dS-space (BdS). It is important to emphasize that the Beltrami coordinates or inhomogeneous projective ones, without the antipodal identification for preserving orientation, play an important role in analogy with the Mink-coordinates in a Mink-space. Namely, in the Beltrami atlas, particles and light signals move along the time-like or null geodesics being straight world-lines with constant coordinate velocities, respectively. Among these systems, the properties are invariant under the fractionally linear transforma...

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