K-Poincaré group

In physics and mathematics, the κ-Poincaré group is a quantum group, obtained by deformation of the Poincaré group into an Hopf algebra.It is generated by the elements $a^mu$ and $\{Lambda^mu\}\_\; u$ with the usual constraint:$eta^\{\; ho\; sigma\}\; \{Lambda^mu\}\_\; ho\; \{Lambda^\; u\}\_sigma\; =\; eta^\{mu\; u\}\; ~,$where $eta^\{mu\; u\}$ is the Minkowskian metric:

The commutation rules reads:
* $[a\_j\; ,a\_0]\; =\; i\; lambda\; a\_j\; ~,\; ;\; [a\_j,a\_k]\; =0\; ,$
* $[a^mu\; ,\; \{Lambda^\; ho\}\_sigma\; ]\; =\; i\; lambda\; left\{\; left(\; \{Lambda^\; ho\}\_0\; -\; \{delta^\; ho\}\_0\; ight)\; \{Lambda^mu\}\_sigma\; -\; left(\; \{Lambda^alpha\}\_sigma\; eta\_\{alpha\; 0\}\; +\; eta\_\{sigma\; 0\}\; ight)\; eta^\{\; ho\; mu\}\; ight\}\; ,$

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

and the commutation rules reads:

*
*

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

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

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

References