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:

• $[P_{\mu },P_{\nu }]=0$
• $[R_j,P_0]=0,[R_j,P_k]=i{\epsilon }_{jkl}{P}_l,[R_j,N_k]=i{\epsilon }_{jkl}{N}_l,[R_j,R_k]=i{\epsilon }_{jkl}{R}_l$
• $[N_j,P_0]=i{P}_j,[N_j,P_k]=i{\delta }_{jk}(\frac{1-e^{-2\lambda P_0}}{2\lambda }+\frac{\lambda }{2}|\overrightarrow{P}{|}^2)-i\lambda P_j{P}_k,[N_j,N_k]=-i{\epsilon }_{jkl}{R}_l$

Where $P_{\mu }$ are the translation generators, $R_j$ the rotations and $N_j$ the boosts. The coproducts are:

• $\mathrm{\Delta }{P}_j=P_j\otimes 1+e^{-\lambda P_0}\otimes P_j,\mathrm{\Delta }{P}_0=P_0\otimes 1+1\otimes P_0$
• $\mathrm{\Delta }{R}_j=R_j\otimes 1+1\otimes R_j$
• $\mathrm{\Delta }{N}_k=N_k\otimes 1+e^{-\lambda P_0}\otimes N_k+i\lambda {\epsilon }_{klm}{P}_l\otimes R_m.$

The antipodes and the counits:

• $S(P_0)=-P_0$
• $S(P_j)=-e^{\lambda P_0}{P}_j$
• $S(R_j)=-R_j$
• $S(N_j)=-e^{\lambda P_0}{N}_j+i\lambda {\epsilon }_{jkl}{e}^{\lambda P_0}{P}_k{R}_l$
• $\epsilon (P_0)=0$
• $\epsilon (P_j)=0$
• $\epsilon (R_j)=0$
• $\epsilon (N_j)=0$

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