K-Poincaré algebra

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


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