In physics and mathematics, the κ-Poincaré algebra, named after Henri Poincaré, is a deformation of the Poincaré algebra into a Hopf algebra. In the bicrossproduct basis, introduced by Majid-Ruegg[1] its commutation rules reads:
![{\displaystyle [P_{\mu },P_{\nu }]=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/737e9f4f0edf9b8e72332c929615b7b632af6702)
![{\displaystyle [R_{j},P_{0}]=0,\;[R_{j},P_{k}]=i\varepsilon _{jkl}P_{l},\;[R_{j},N_{k}]=i\varepsilon _{jkl}N_{l},\;[R_{j},R_{k}]=i\varepsilon _{jkl}R_{l}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4196aaf2d4b742f3b4c1991e5db78a54adaaef88)
![{\displaystyle [N_{j},P_{0}]=iP_{j},\;[N_{j},P_{k}]=i\delta _{jk}\left({\frac {1-e^{-2\lambda P_{0}}}{2\lambda }}+{\frac {\lambda }{2}}|{\vec {P}}|^{2}\right)-i\lambda P_{j}P_{k},\;[N_{j},N_{k}]=-i\varepsilon _{jkl}R_{l}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/72efaa65139f14167f6c908fd96dc5dc1ec1e846)
Where
are the translation generators,
the rotations and
the boosts.
The coproducts are:
![{\displaystyle \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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bbbc54f199fd8f4986c55a83188004733cd777f8)
![{\displaystyle \Delta R_{j}=R_{j}\otimes 1+1\otimes R_{j}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/45afb0a9960ca1dd7e1af87cce1d957d3cf7107e)
![{\displaystyle \Delta N_{k}=N_{k}\otimes 1+e^{-\lambda P_{0}}\otimes N_{k}+i\lambda \varepsilon _{klm}P_{l}\otimes R_{m}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e8db25df420c112e9a5790a090d106d4aebe0e3e)
The antipodes and the counits:
![{\displaystyle S(P_{0})=-P_{0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b9f2f0cdef9f23c0e2df491d3109c9202fcd9fff)
![{\displaystyle S(P_{j})=-e^{\lambda P_{0}}P_{j}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b62c65f3908a4ad133dd0253cff73eacb6b5f94d)
![{\displaystyle S(R_{j})=-R_{j}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3267a20cbf3bdea534cec21efeebe545b0d62024)
![{\displaystyle S(N_{j})=-e^{\lambda P_{0}}N_{j}+i\lambda \varepsilon _{jkl}e^{\lambda P_{0}}P_{k}R_{l}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/db6d05094d527a59e6ee7f016eeb70a6a619ac65)
![{\displaystyle \varepsilon (P_{0})=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b2285bbe4e5bbf4afe08415890885f3103e541cf)
![{\displaystyle \varepsilon (P_{j})=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0f63b9f263cb3d9382718cadd8e2c4651af1bbdd)
![{\displaystyle \varepsilon (R_{j})=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dae8dc797c24c505081f0bc3b6b62452108467e9)
![{\displaystyle \varepsilon (N_{j})=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/07e792892c6ded985e9567895c50fcfaa7f80774)
The κ-Poincaré algebra is the dual Hopf algebra to the κ-Poincaré group, and can be interpreted as its “infinitesimal” version.
References