Different Bases of, к-Deformed Poincaré Algebra

Abstract

The contraction of U q (0(3,2)) |q| = 1 [1, 2] or q real [3]) provided first quantum deformations U к (P4) of D = 4 Poincare algebra P4≡(M µv ,P µ ,) with κ describing the mass-like deformation parameter 1). These so-called ic-deformations are considered in the class of noncommutative and noncocommutative Hopf algebras [4–6]with modified classical coalgebra sector. It should be stressed that the choice of ten generators obtained in [3] is not unique: one can distinguish at least two other bases, with quite interesting properties: i) The bicrossproduct basis, obtained in [7]. In such a basis the quantum algebra U к P(4) can be written in the form 2) $${\mathcal{U}_\kappa }\left( {{\mathcal{P}_4}} \right) = \mathcal{U}\left( {O\left( {3,1} \right)} \right)\blacktriangleright \triangleleft T_4^\kappa$$ where —U(0(3,1)) describes the Hopf algebra generated by classical Lorentz, generators, with commutative coproducts \(T_4^\kappa\)describes the K-deformed Hopf algebra of fourmomenta, with commuting generators in algebra sector and K-deformed coalgebra relations. ii) The classical Poincare algebra basis, obtained in [11]3). In such a framework the algebra is a standard Lie algebra, but the coproducts are very complicated noncocommutative expressions.