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\(\kappa\)-deformed covariant quantum phase spaces as Hopf algebroids. (English) Zbl 1364.81170
Summary: We consider the general \(D = 4\) \((10 + 10)\)-dimensional \(\kappa\)-deformed quantum phase space as given by Heisenberg double \(\mathcal{H}\) of \(D = 4\) \(\kappa\)-deformed Poincaré-Hopf algebra \(\mathbb{H}\). The standard \((4 + 4)\)-dimensional \(\kappa\)-deformed covariant quantum phase space spanned by \(\kappa\)-deformed Minkowski coordinates and commuting momenta generators \((\hat{x}_\mu, \hat{p}_\mu)\) is obtained as the subalgebra of \(\mathcal{H}\). We study further the property that Heisenberg double defines particular quantum spaces with Hopf algebroid structure. We calculate by using purely algebraic methods the explicit Hopf algebroid structure of standard \(\kappa\)-deformed quantum covariant phase space in Majid-Ruegg bicrossproduct basis. The coproducts for Hopf algebroids are not unique, determined modulo the coproduct gauge freedom. Finally we consider the interpretation of the algebraic description of quantum phase spaces as Hopf algebroids.

MSC:
81S30 Phase-space methods including Wigner distributions, etc. applied to problems in quantum mechanics
81R05 Finite-dimensional groups and algebras motivated by physics and their representations
16T05 Hopf algebras and their applications
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