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 $${\Lambda^\mu}_\nu$$ with the usual constraint:



\eta^{\rho \sigma} {\Lambda^\mu}_\rho {\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:
 * $$[a_j ,a_0] = i \lambda a_j ~, \; [a_j,a_k]=0 $$
 * $$ [a^\mu, {\Lambda^\rho}_\sigma ]  = i \lambda \left\{ \left( {\Lambda^\rho}_0 - {\delta^\rho}_0 \right) {\Lambda^\mu}_\sigma - \left( {\Lambda^\alpha}_\sigma \eta_{\alpha 0}  + \eta_{\sigma 0} \right) \eta^{\rho \mu} \right\} $$

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


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

and the commutation rules reads:


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

The coproducts are classical, and encode the group composition law:
 * $$\Delta a^\mu = {\Lambda^\mu}_\nu \otimes a^\nu + a^\mu \otimes 1 $$
 * $$\Delta {\Lambda^\mu}_\nu = {\Lambda^\mu}_\rho \otimes {\Lambda^\rho}_\nu $$

Also the antipodes and the counits are classical, and represent the group inversion law and the map to the identity:
 * $$S(a^\mu) = - {(\Lambda^{-1})^\mu}_\nu a^\nu $$
 * $$S({\Lambda^\mu}_\nu) = {(\Lambda^{-1})^\mu}_\nu = {\Lambda_\nu}^\mu $$
 * $$\varepsilon (a^\mu) = 0$$
 * $$\varepsilon ({\Lambda^\mu}_\nu) ={\delta^\mu}_\nu $$

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