Quantum Poincare group related to the kappa -Poincare algebra

On κ-deformation and triangular quasibialgebra structure

“…We have verified using FORM that φ 4 exists and is unique; its actual expression is very lengthy and we omit it. Note that, as expected, the first non-vanishing term is nothing but φ 2 = 1 12 M µν ∧ P µ ∧ P ν , which is the (classically-Poincaré invariant) source term that appears in the MCYBE obeyed by the classical r-matrix [34,36].…”

“…which is the classical r-matrix associated to U κ (P), and has been known since the work of [34]; see also [35,36]. The solution r 1 is unique up to the addition of terms that commute with E ∨1, P i ∨1, N i ∨1 to leading order; in other words, terms that are classically 4 Poincaré invariant.…”

Covariant particle exchange for κ-deformed theories in dimensions

“…5 This has the merit of concreteness, but it obscures the symmetry implicit in τ 2 = id. We want to show here that, in the 1+1-dimensional case at least, the claim is true exactly rather than just perturbatively, so we approach the problem from a point of view in which this symmetry is manifest from the start.…”

“…From the dual point of view, in the Hopf-algebraic sense [4], there is a κ-deformation of the algebra of functions on the Poincaré group [5].…”

Poisson Structures on the Poincaré Group

“…the notion of a Poisson-Lie group [3] (see also references in [2]) is simple, clear and general, whereas the notion of a quantum deformation is rather difficult, varies from author to author, and can be different for different types of groups (for instance, a separate definition for semidirect products), 2. calculations of Poisson structures are technically much easier (classical Yang-Baxter equation is quadratic whereas the quantum one is cubic) and have often a direct Lie-algebraic meaning, 3. in some cases it is easy to pass from a Poisson-Lie group to the corresponding quantum group [4]: the classical r-matrix can be used to (a) construct all remaining objects, (b) denote the deformation (convenient when communicating with other people), 4. it is easier to check whether the Poisson-Lie group is non-complete than to check if the corresponding quantum deformation (on the Hopf * -algebra level) can be formulated on the C * -algebra level [5].…”