Abstract
Realizations of $\kappa$-Minkowski space linear in momenta are studied for time-, space- and light-like deformations. We construct and classify all such linear realizations and express them in terms of $\mathfrak{gl}(n)$ generators. There are three one-parameter families of linear realizations for time-like and space-like deformations, while for light-like deformations, there are only four linear realizations. The relation between deformed Heisenberg algebra, star product, coproduct of momenta and twist operator is presented. It is proved that for each linear realization there exists Drinfeld twist satisfying normalization and cocycle conditions. $\kappa$-deformed $\mathfrak{igl}(n)$-Hopf algebras are presented for all cases. The $\kappa$-Poincar\'e-Weyl and $\kappa$-Poincar\'e-Hopf algebras are discussed. Left-right dual $\kappa$-Minkowski algebra is constructed from the transposed twists. The corresponding realizations are nonlinear. All known Drinfeld twists related to $\kappa$-Minkowski space are obtained from our construction. Finally, some physical applications are discussed.
Highlights
It is proved that for each linear realization there exists a Drinfeld twist satisfying normalization and cocycle conditions. κ-Deformed igl(n)-Hopf algebras are presented for all cases
Κ-Deformed Poincaré symmetry is algebraically described by the κ-Poincaré–Hopf algebra and is an example of deformed relativistic symmetry that can possibly describe the physical reality at the Planck scale. κ is the deformation parameter, usually interpreted as the Planck mass or some quantum gravity scale
It was shown that quantum field theory with κ-Poincaré symmetry emerges in a certain limit of quantum gravity coupled to matter fields after integrating out the gravitational/topological degrees of freedom [11,12,13,14,15]
Summary
Commutative coordinates xμ and momenta pμ generate an undeformed Heisenberg algebra H given by [xμ, xν] = 0,. To Ain the previous section, commutative coordinates xμ generate an enveloping algebra A, which is subalgebra of undeformed Heisenberg algebra, i.e. A ⊂ H. Momenta pμ generate algebra T , which is a subalgebra of the undeformed Heisenberg algebra, i.e. T ⊂ H. The undeformed Heisenberg algebra is, symbolically, H = AT. We are looking for linear realizations of κ-Minkowski space, that is, the realizations where the function φαμ( p) is linear in pμ. They can be written in the form xμ = xμ + lμ,. It follows that lμ satisfies the same commutation relations as xμ:.
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