K-Poincaré Algebra

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 its commutation rules reads:

* _\mu, P_\nu= 0
* _j , P_0= 0, \; _j , P_k= i \varepsilon_ P_l, \; _j , N_k= i \varepsilon_ N_l, \; _j , R_k= i \varepsilon_ R_l
* _j , P_0= i P_j, \; _j , P_k= i \delta_ \left( \frac + \frac , \vec, ^2 \right) - i \lambda P_j P_k, \; _j,N_k= -i \varepsilon_ R_l

Where $P_\mu$ are the translation generators, $R_j$ the rotations and $N_j$ the boosts.
The coproducts are:
* $\Delta P_j = P_j \otimes 1 + e^ \otimes P_j ~, \qquad \Delta P_0 = P_0 \otimes 1 + 1 \otimes P_0$
* $\Delta R_j = R_j \otimes 1 + 1 \otimes R_j$
* $\Delta N_k = N_k \otimes 1 + e^ \otimes N_k + i \lambda \varepsilon_ P_l \otimes R_m .$

The antipodes and the counits:
* $S(P_0) = - P_0$
* $S(P_j) = -e^ P_j$
* $S(R_j) = - R_j$
* $S(N_j) = -e^N_j +i \lambda \varepsilon_ e^ P_k R_l$
* $\varepsilon(P_0) = 0$
* $\varepsilon(P_j) = 0$
* $\varepsilon(R_j) = 0$
* $\varepsilon(N_j) = 0$

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

References

Hopf algebras
Mathematical physics

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