Noncommutative Differential Calculus on the Κ-minkowski Space

Following the construction of the κ-Minkowski space from the bicrossproduct structure of the κ-Poincare group, we investigate possible differential calculi on this noncommutative space. We discuss then the action of the Lorentz quantum algebra and prove that there are no 4D bicovariant differential calculi, which are Lorentz covariant. We show, however, that there exist a fivedimensional differential calculus, which satisfies both requirements. We study also a toy example of 2D κ-Minkowski space and and we briefly discuss the main properties of its differential calculi. Partially supported by KBN Grant 2P 302 103 06 Permanent address: Department of Field Theory, Institute of Physics, Jagiellonian University, Reymonta 4, 30-059 Kraków, Poland, e-mail: [email protected] 1 The κ-Poincare algebra The κ-Poincare algebra has been introduced [1, 2] and studied extensively [3, 4, 5, 6, 7] as one of possible Hopf algebra deformations of the standard Poincare algebra. The momenta and the rotation generators remain unchanged and the deformation occurs only in the boost sector and the coproduct structure. Recently, it was shown [8] that the κ-Poincare has a structure of a Hopf algebra extension of (classical) U(so(1, 3)) by the Hopf algebra of (deformed) translations T : [Pμ, Pν ] = 0, (1) ∆P0 = P0⊗ 1 + 1⊗P0, (2) ∆Pi = Pi ⊗ 1 + e − P0 κ ⊗Pi. (3) The commutation relations of the Lorentz algebra generators, rotations Mi and boosts Ni are the standard ones: [Mi,Mj ] = ǫijkMk, [Mi, Nj] = ǫijkNk, [Ni, Nj] = −ǫijkMk, (4) whereas the cross relations and the coproduct structure of the Lorentz part are deformed: [P0,Mi] = 0, [Pi,Mj ] = ǫijkPk (5) [P0, Ni] = −Pi, [Pi, Nj] = −δij( κ 2 (1− e 2P0 κ ) + 1 2κ ~ P ) + 1 κ PiPj, (6) ∆Ni = Ni ⊗ 1 + e − P0 κ ⊗Ni + ǫijk κ Pj ⊗Mk, ∆Mi = Mi ⊗ 1 + 1⊗Mi. (7) The classical Poincare algebra is obtained in the limit κ → ∞.