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[1] its commutation rules reads:

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

Where ${\displaystyle P_{\mu }}$ are the translation generators, ${\displaystyle R_{j}}$ the rotations and ${\displaystyle N_{j}}$ the boosts. The coproducts are:

• ${\displaystyle \Delta P_{j}=P_{j}\otimes 1+e^{-\lambda P_{0}}\otimes P_{j}~,\qquad \Delta P_{0}=P_{0}\otimes 1+1\otimes P_{0}}$
• ${\displaystyle \Delta R_{j}=R_{j}\otimes 1+1\otimes R_{j}}$
• ${\displaystyle \Delta N_{k}=N_{k}\otimes 1+e^{-\lambda P_{0}}\otimes N_{k}+i\lambda \varepsilon _{klm}P_{l}\otimes R_{m}.}$

The antipodes and the counits:

• ${\displaystyle S(P_{0})=-P_{0}}$
• ${\displaystyle S(P_{j})=-e^{\lambda P_{0}}P_{j}}$
• ${\displaystyle S(R_{j})=-R_{j}}$
• ${\displaystyle S(N_{j})=-e^{\lambda P_{0}}N_{j}+i\lambda \varepsilon _{jkl}e^{\lambda P_{0}}P_{k}R_{l}}$
• ${\displaystyle \varepsilon (P_{0})=0}$
• ${\displaystyle \varepsilon (P_{j})=0}$
• ${\displaystyle \varepsilon (R_{j})=0}$
• ${\displaystyle \varepsilon (N_{j})=0}$

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