K-Poincaré algebra

In physics and mathematics, the κ-Poincaré algebra, named after Henri Poincaré, is a deformation of the Poincaré algebra into a Hopf algebra. In the bicrossproduct basis, introduced by Majid-Ruegg its commutation rules reads:


 * $$[P_\mu, P_\nu] = 0 $$
 * $$ [R_j, P_0] = 0, \; [R_j , P_k] = i \varepsilon_{jkl} P_l, \; [R_j , N_k] = i \varepsilon_{jkl} N_l, \; [R_j , R_k] = i \varepsilon_{jkl} R_l$$
 * $$[N_j, P_0] = i P_j, \;[N_j , P_k] = i \delta_{jk} \left( \frac{1 - e^{- 2 \lambda P_0}}{2 \lambda}  + \frac{ \lambda }{2}  |\vec{P}|^2 \right) - i \lambda P_j P_k, \; [N_j,N_k] = -i \varepsilon_{jkl} R_l$$

Where $$P_\mu$$ are the translation generators, $$R_j$$ the rotations and $$N_j$$ the boosts. The coproducts are:
 * $$\Delta P_j = P_j \otimes 1 +  e^{- \lambda P_0} \otimes P_j ~, \qquad \Delta P_0  = P_0 \otimes 1 + 1 \otimes P_0$$
 * $$\Delta R_j = R_j \otimes 1 +  1 \otimes R_j$$
 * $$\Delta N_k =  N_k \otimes 1 + e^{-\lambda P_0} \otimes N_k  + i \lambda \varepsilon_{klm}  P_l \otimes R_m .$$

The antipodes and the counits:
 * $$S(P_0) = - P_0$$
 * $$S(P_j) = -e^{\lambda P_0} P_j$$
 * $$S(R_j) = - R_j$$
 * $$S(N_j) =  -e^{\lambda P_0}N_j +i \lambda \varepsilon_{jkl} e^{\lambda P_0} P_k R_l$$
 * $$\varepsilon(P_0) = 0$$
 * $$\varepsilon(P_j) = 0$$
 * $$\varepsilon(R_j) = 0$$
 * $$\varepsilon(N_j) = 0$$

The κ-Poincaré algebra is the dual Hopf algebra to the κ-Poincaré group, and can be interpreted as its “infinitesimal” version.