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Quantum twist-deformed \(D = 4\) phase spaces with spin sector and Hopf algebroid structures. (English) Zbl 1406.81050
Summary: 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 \(\widehat{\mathbb{G}}\) provides by the Heisenberg double algebra \(\mathcal{HD} = \mathbb{H} \rtimes \widehat{\mathbb{G}}\), and second, denoted by \(\widetilde{\mathcal{H}}^{(10, 10)}\), 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.

81R60 Noncommutative geometry in quantum theory
16T05 Hopf algebras and their applications
81R05 Finite-dimensional groups and algebras motivated by physics and their representations
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