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Poincaré group
K-Poincaré group

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Poincaré group

The Poincaré group, named after Henri Poincaré (1906), was first defined by Minkowski (1908) being the group of Minkowski spacetime isometries. It is a ten-generator non-abelian Lie group of fundamental importance in physics.

Overview

A Minkowski spacetime isometry has the property that the interval between events is left invariant. For example, if everything was postponed by two hours, including the two events and the path you took to go from one to the other, then the time interval between the events recorded by a stop-watch you carried with you would be the same. Or if everything was shifted five miles to the west, or turned 60 degrees to the right, you would also see no change in the interval. It turns out that the proper length of an object is also unaffected by such a shift. A time or space reversal (a reflection) is also an isometry of this group.

In Minkowski space (i.e. ignoring the effects of gravity), there are ten degrees of freedom of the isometries, which may be thought of as translation through time or space (four degrees, one per dimension); reflection through a plane (three degrees, the freedom in orientation of this plane); or a "boost" in any of the three spatial directions (three degrees). Composition of transformations is the operator of the Poincaré group, with proper rotations being produced as the composition of an even number of reflections.

This page contains text from Wikipedia, the Free Encyclopedia - https://wn.com/Poincaré_group

K-Poincaré group

In physics and mathematics, the κ-Poincaré group, named after Henri Poincaré, is a quantum group, obtained by deformation of the Poincaré group into an Hopf algebra. It is generated by the elements $a^\mu$ and ${\Lambda^\mu}_\nu$ with the usual constraint:

where $\eta^{\mu \nu}$ is the Minkowskian metric:

The commutation rules reads:

[a_j ,a_0] = i \lambda a_j ~, \; [a_j,a_k]=0 \,

[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\} \,

In the (1 + 1)-dimensional case the commutation rules between $a^\mu$ and ${\Lambda^\mu}_\nu$ are particularly simple. The Lorentz generator in this case is:

and the commutation rules reads:

[ 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) \,

[ 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) \,

The coproducts are classical, and encode the group composition law:

\Delta a^\mu = {\Lambda^\mu}_\nu \otimes a^\nu + a^\mu \otimes 1 \,

\Delta {\Lambda^\mu}_\nu = {\Lambda^\mu}_\rho \otimes {\Lambda^\rho}_\nu \,

Also the antipodes and the counits are classical, and represent the group inversion law and the map to the identity:

S(a^\mu) = - {(\Lambda^{-1})^\mu}_\nu a^\nu \,

S({\Lambda^\mu}_\nu) = {(\Lambda^{-1})^\mu}_\nu = {\Lambda_\nu}^\mu \,

\varepsilon (a^\mu) = 0

\varepsilon ({\Lambda^\mu}_\nu) ={\delta^\mu}_\nu \,

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

References

This page contains text from Wikipedia, the Free Encyclopedia - https://wn.com/K-Poincaré_group

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