K-Poincaré algebra

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

• $[P_{\mu },P_{\nu }]=0\,$
• $[R_{j},P_{0}]=0,\;[R_{j},P_{k}]=i\varepsilon _{jkl}P_{l},\;[R_{j},N_{k}]=i\varepsilon _{jkl}N_{l},\;[R_{j},R_{k}]=i\varepsilon _{jkl}R_{l}\,$
• $[N_{j},P_{0}]=iP_{j},\;[N_{j},P_{k}]=i\delta _{jk}\left({\frac {1-e^{-2\lambda P_{0}}}{2\lambda }}+{\frac {\lambda }{2}}|{\vec {P}}|^{2}\right)-i\lambda P_{j}P_{k},\;[N_{j},N_{k}]=-i\varepsilon _{jkl}R_{l}\,$

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.