K-Poincaré group

In physics and mathematics, the κ-Poincaré group, named after Henri Poincaré, is a quantum group, obtained by deformation of the Poincaré group into a Hopf algebra. It is generated by the elements $a^{\mu }$ and ${{\mathrm{\Lambda }}^{\mu }}_{\nu }$ with the usual constraint:

${\eta }^{\rho \sigma }{{\mathrm{\Lambda }}^{\mu }}_{\rho }{{\mathrm{\Lambda }}^{\nu }}_{\sigma }={\eta }^{\mu \nu },$

where ${\eta }^{\mu \nu }$ is the Minkowskian metric:

${\eta }^{\mu \nu }=\left(\begin{array}{cccc}-1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right).$

The commutation rules reads:

• $[a_j,a_0]=i\lambda a_j,[a_j,a_k]=0$
• $[a^{\mu },{{\mathrm{\Lambda }}^{\rho }}_{\sigma }]=i\lambda \{({{\mathrm{\Lambda }}^{\rho }}_0-{{\delta }^{\rho }}_0){{\mathrm{\Lambda }}^{\mu }}_{\sigma }-({{\mathrm{\Lambda }}^{\alpha }}_{\sigma }{\eta }_{\alpha 0}+{\eta }_{\sigma 0}){\eta }^{\rho \mu }\}$

In the (1 + 1)-dimensional case the commutation rules between $a^{\mu }$ and ${{\mathrm{\Lambda }}^{\mu }}_{\nu }$ are particularly simple. The Lorentz generator in this case is:

${{\mathrm{\Lambda }}^{\mu }}_{\nu }=\left(\begin{array}{cc}\cosh \tau & \sinh \tau \\ \sinh \tau & \cosh \tau \end{array}\right)$

and the commutation rules reads:

• $[a_0,\left(\begin{array}{c}\cosh \tau \\ \sinh \tau \end{array}\right)]=i\lambda \sinh \tau \left(\begin{array}{c}\sinh \tau \\ \cosh \tau \end{array}\right)$
• $[a_1,\left(\begin{array}{c}\cosh \tau \\ \sinh \tau \end{array}\right)]=i\lambda (1-\cosh \tau )\left(\begin{array}{c}\sinh \tau \\ \cosh \tau \end{array}\right)$

The coproducts are classical, and encode the group composition law:

• $\mathrm{\Delta }{a}^{\mu }={{\mathrm{\Lambda }}^{\mu }}_{\nu }\otimes a^{\nu }+a^{\mu }\otimes 1$
• $\mathrm{\Delta }{{\mathrm{\Lambda }}^{\mu }}_{\nu }={{\mathrm{\Lambda }}^{\mu }}_{\rho }\otimes {{\mathrm{\Lambda }}^{\rho }}_{\nu }$

Also the antipodes and the counits are classical, and represent the group inversion law and the map to the identity:

• $S(a^{\mu })=-{({\mathrm{\Lambda }}^{-1}{)}^{\mu }}_{\nu }{a}^{\nu }$
• $S({{\mathrm{\Lambda }}^{\mu }}_{\nu })={({\mathrm{\Lambda }}^{-1}{)}^{\mu }}_{\nu }={{\mathrm{\Lambda }}_{\nu }}^{\mu }$
• $\epsilon (a^{\mu })=0$
• $\epsilon ({{\mathrm{\Lambda }}^{\mu }}_{\nu })={{\delta }^{\mu }}_{\nu }$

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