The quantum phase space described by Heisenberg algebra possesses undeformed Hopf algebroid structure. The κ-deformed phase space with noncommutative coordinates is realized in terms of undeformed quantum phase space. There are infinitely many such realizations related by similarity transformations. For a given realization, we construct corresponding coproducts of commutative coordinates and momenta (bialgebroid structure). The κ-deformed phase space has twisted Hopf algebroid structure. General method for the construction of twist operator (satisfying cocycle and normalization condition) corresponding to deformed coalgebra structure is presented. Specially, twist for natural realization (classical basis) of κ-Minkowski space-time is presented. The cocycle condition, κ-Poincaré algebra and R-matrix are discussed. Twist operators in arbitrary realizations are constructed from the twist in the given realization using similarity transformations. Some examples are presented. The important physical applications of twists, realizations, R-matrix and Hopf algebroid structure are discussed. Int. J. Mod. Phys. A 2014.29. Downloaded from www.worldscientific.com by UNIVERSITY OF TORONTO on 02/03/15. For personal use only. 1450022-3 Int. J. Mod. Phys. A 2014.29. Downloaded from www.worldscientific.com by UNIVERSITY OF TORONTO on 02/03/15. For personal use only. 1450022-4 Int. J. Mod. Phys. A 2014.29. Downloaded from www.worldscientific.com by UNIVERSITY OF TORONTO on 02/03/15. For personal use only. 1450022-5 Int. J. Mod. Phys. A 2014.29. Downloaded from www.worldscientific.com by UNIVERSITY OF TORONTO on 02/03/15. For personal use only. 1450022-6 Int. J. Mod. Phys. A 2014.29. Downloaded from www.worldscientific.com by UNIVERSITY OF TORONTO on 02/03/15. For personal use only. multiplication map defined by m ⋆1 . The property of right ideal J is m ⋆ (J ⊲ (f ⊗ g)) = 0. For the right ideals J 0 and J , we have J 0 (H ⊗ H) = J 0 , J (H ⊗ H) = J , J = F J 0 F −1 = F J 0 , J J 0 ⊂ J , J 0 J ⊂ J 0 .

Classification of differential forms on κ-Minkowski space, particularly, the classification of all bicovariant differential calculi of classical dimension is presented. By imposing super-Jacobi identities we derive all possible differential algebras compatible with the κ-Minkowski algebra for time-like, space-like and light-like deformations. Embedding into the super-Heisenberg algebra is constructed using non-commutative (NC) coordinates and one-forms. Particularly, a class of differential calculi with an undeformed exterior derivative and one-forms is considered. Corresponding NC differential calculi are elaborated. Related class of new Drinfeld twists is proposed. It contains twist leading to κ-Poincaré Hopf algebra for light-like deformation. Corresponding super-algebra and deformed super-Hopf algebras, as well as the symmetries of differential algebras are presented and elaborated. Using the NC differential calculus, we analyze NC field theory, modified dispersion relations, and discuss further physical applications.

We propose a generalized description for the κ-Poincaré-Hopf algebra as a symmetry quantum group of underlying κ-Minkowski spacetime. We investigate all the possible implementations of (deformed) Lorentz algebras which are compatible with the given choice of κ-Minkowski algebra realization. For the given realization of κ-Minkowski spacetime there is a unique κ-Poincaré-Hopf algebra with undeformed Lorentz algebra. We have constructed a three-parameter family of deformed Lorentz generators with κ-Poincaré algebras which are related to κ-Poincaré-Hopf algebra with undeformed Lorentz algebra. Known bases of κ-Poincaré-Hopf algebra are obtained as special cases. Also deformation of igl(4) Hopf algebra compatible with the κ-Minkowski spacetime is presented. Some physical applications are briefly discussed. 1 domagoj.kovacevic@fer.hr 2 meljanac@irb.hr 3 pachol@raunvis.hi.is 4 rina.

We deform a phase space ( Heisenberg algebra and corresponding coalgebra) by twist. We present undeformed and deformed tensor identities that are crucial in our construction. Coalgebras for the generalized Poincaré algebras have been constructed. The exact universal R-matrix for the deformed Heisenberg (co)algebra is found. We show, up to the third order in the deformation parameter, that in the case of κ-Poincaré Hopf algebra this R-matrix can be expressed in terms of Poincaré generators only. This implies that the states of any number of identical particles can be defined in a κ-covariant way.