K-Poincaré algebra

In physics and mathematics, the κ-Poincaré algebra is a deformation of the Poincaré algebra into an Hopf algebra. In the bicrossproduct basis, introduced by Majid-Ruegg [Majid-Ruegg, Phys. Lett. B 334 (1994) 348, ArXiv: [http://arxiv.org/abs/hep-th/9405107 hep-th/9405107] ] its commutation rules reads:

* $[P\_mu,\; P\_\; u]\; =\; 0\; ,$

* $[R\_j\; ,\; P\_0]\; =\; 0,\; ;\; [R\_j\; ,\; P\_k]\; =\; i\; varepsilon\_\{jkl\}\; P\_l,\; ;\; [R\_j\; ,\; P\_0]\; =\; 0,\; ;\; [R\_j\; ,\; N\_k]\; =\; i\; varepsilon\_\{jkl\}\; N\_l,$

* $[N\_j\; ,\; P\_0]\; =\; i\; P\_j,\; ;\; [N\_j\; ,\; P\_k]\; =\; i\; delta\_\{jk\}\; left(\; frac\{1\; -\; e^\{-\; 2\; lambda\; P\_0\{2\; lambda\}\; +\; frac\{\; lambda\; \}\{2\}\; |vec\{P\}|^2\; ight),\; ;\; [N\_j,N\_k]\; =\; -i\; varepsilon\_\{jkl\}\; R\_l,$

Where $P\_mu$ are the translation generators, $R\_j$ the rotations and $N\_j$ the boosts.The coproducts are:
* $Delta\; P\_j\; =\; P\_j\; otimes\; 1\; +\; e^\{-\; lambda\; P\_0\}\; otimes\; P\_j\; ~,\; qquad\; Delta\; P\_0\; =\; P\_0\; otimes\; 1\; +\; 1\; otimes\; P\_0,$
* $Delta\; R\_j\; =\; R\_j\; otimes\; 1\; +\; 1\; otimes\; R\_j,$
* $Delta\; N\_k\; =\; N\_k\; otimes\; 1\; +\; e^\{-lambda\; P\_0\}\; otimes\; N\_k\; +\; i\; lambda\; varepsilon\_\{klm\}\; P\_l\; otimes\; R\_m\; .$

The antipodes and the counits:
* $S(P\_0)\; =\; -\; P\_0,$
* $S(P\_j)\; =\; -e^\{lambda\; P\_0\}\; P\_j,$
* $S(R\_j)\; =\; -\; R\_j,$
* $S(N\_j)\; =\; -e^\{lambda\; P\_0\}N\_j\; +\; lambda\; varepsilon\_\{jkl\}\; e^\{lambda\; P\_0\}\; P\_k\; R\_l,$

* $varepsilon(P\_0)\; =\; 0,$
* $varepsilon(P\_j)\; =\; 0,$
* $varepsilon(R\_j)\; =\; 0,$
* $varepsilon(N\_j)\; =\; 0,$

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

References