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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
and
with the usual constraint:
![{\displaystyle \eta ^{\rho \sigma }{\Lambda ^{\mu }}_{\rho }{\Lambda ^{\nu }}_{\sigma }=\eta ^{\mu \nu }~,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/072ce5ea8989cdbae8f1cb1994e078daa9a03b1f)
where
is the Minkowskian metric:
![{\displaystyle \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)~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/69bc86f166a9134ee30e52e89d97df44e4e91240)
The commutation rules reads:
![{\displaystyle [a_{j},a_{0}]=i\lambda a_{j}~,\;[a_{j},a_{k}]=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3fa0e644380cd0793a1b5f5561be507b5e362f92)
![{\displaystyle [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\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee5d32c295acaec499440c3284c084d22119bdcf)
In the (1 + 1)-dimensional case the commutation rules between
and
are particularly simple. The Lorentz generator in this case is:
![{\displaystyle {\Lambda ^{\mu }}_{\nu }=\left({\begin{array}{cc}\cosh \tau &\sinh \tau \\\sinh \tau &\cosh \tau \end{array}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/954df013f9bea4181532e22060200be07eadf1a5)
and the commutation rules reads:
![{\displaystyle [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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6b14f84dc7c0c6f339c52eccd881bcaa9d9f385b)
![{\displaystyle [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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec6e1d13266bfcd5cde7822ec98d223c970734f7)
The coproducts are classical, and encode the group composition law:
![{\displaystyle \Delta a^{\mu }={\Lambda ^{\mu }}_{\nu }\otimes a^{\nu }+a^{\mu }\otimes 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1ae822a6de78755a51ffa01a241d2ae73f5cb197)
![{\displaystyle \Delta {\Lambda ^{\mu }}_{\nu }={\Lambda ^{\mu }}_{\rho }\otimes {\Lambda ^{\rho }}_{\nu }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b6dbf5febf1642654891a3d6548f2cc22ee7d780)
Also the antipodes and the counits are classical, and represent the group inversion law and the map to the identity:
![{\displaystyle S(a^{\mu })=-{(\Lambda ^{-1})^{\mu }}_{\nu }a^{\nu }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2decfab768f96ab8e683239fe258bcb4d983c305)
![{\displaystyle S({\Lambda ^{\mu }}_{\nu })={(\Lambda ^{-1})^{\mu }}_{\nu }={\Lambda _{\nu }}^{\mu }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/176a4fe5a00e928b46f184c65cf46e1a97e9b375)
![{\displaystyle \varepsilon (a^{\mu })=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d710e26cb1c8b78cc5cd1aacf0bcfa5f68146e4c)
![{\displaystyle \varepsilon ({\Lambda ^{\mu }}_{\nu })={\delta ^{\mu }}_{\nu }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e0156d409f72736ea72c8920d2c007a24ee25d6d)
The κ-Poincaré group is the dual Hopf algebra to the K-Poincaré algebra, and can be interpreted as its “finite” version.