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The energy-momentum conservation law in two-particle system for twist-deformed Galilei Hopf algebras

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 Added by Marcin Daszkiewicz
 Publication date 2019
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and research's language is English




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In this article we discus the energy-momentum conservation principle for two-particle system in the case of canonically and Lie-algebraically twist-deformed Galilei Hopf algebra. Particularly, we provide consistent with the coproducts energy and momentum addition law as well as its symmetric with respect the exchange of particles counterpart. Besides, we show that the vanishing of total fourmomentum for two Lie-algebraically deformed kinematical models leads to the discret values of energies and momenta only in the case of the symmetrized addition rules.

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We consider the generalized (10+10)-dimensional D=4 quantum phase spaces containing translational and Lorentz spin sectors associated with the dual pair of twist-quantized Poincare Hopf algebra $mathbb{H}$ and quantum Poincare Hopf group $widehat{mathbb{G}}$. Two Hopf algebroid structures of generalized phase spaces with spin sector will be investigated: first one $% mathcal{H}^{(10,10)}$ describing dynamics on quantum group algebra $% widehat{mathbb{G}}$ provided by the Heisenberg double algebra $mathcal{HD=% }mathbb{H}rtimes widehat{mathbb{G}}$, and second, denoted by $mathcal{% tilde{H}}^{(10,10)}$, describing twisted Hopf algebroid with base space containing twisted noncommutative Minkowski space $hat{x}_{mu }$. We obtain the first explicit example of Hopf algebroid structure of relativistic quantum phase space which contains quantum-deformed Lorentz spin sector.
The Minkowski spacetime quantum Clifford algebra structure associated with the conformal group and the Clifford-Hopf alternative k-deformed quantum Poincare algebra is investigated in the Atiyah-Bott-Shapiro mod 8 theorem context. The resulting algebra is equivalent to the deformed anti-de Sitter algebra U_q(so(3,2)), when the associated Clifford-Hopf algebra is taken into account, together with the associated quantum Clifford algebra and a (not braided) deformation of the periodicity Atiyah-Bott-Shapiro theorem.
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