K-Poincaré group について

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K-Poincaré group ：ウィキペディア英語版

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

_\nu

with the usual constraint:
:

\eta^ _\rho _\sigma = \eta^ ~,

where

\eta^

\eta^ = \left(\begin -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end \right) ~.

The commutation rules reads:
*

(,a_0) = i \lambda a_j ~, \; ()=0 \,

) = i \lambda \left\_0 \right) _\sigma - \left( _\sigma \eta_ + \eta_ \right) \eta^ \right\} \,

In the (1 + 1)-dimensional case the commutation rules between

a^\mu

and

_\nu

are particularly simple. The Lorentz generator in this case is:
:

_\nu = \left( \begin \cosh \tau  & \sinh \tau \\ \sinh \tau & \cosh \tau \end \right) \,

and the commutation rules reads:
*

) = i \lambda ~ \sinh \tau \left( \begin \sinh \tau \\ \cosh \tau \end \right) \,

) = i \lambda \left( 1- \cosh \tau \right) \left( \begin \sinh \tau \\ \cosh \tau \end \right) \,

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

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

\Delta _\nu  = _\rho \otimes _\nu \,

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

S(a^\mu) = - _\nu a^\nu \,

S(_\nu) = _\nu = ^\mu \,

\varepsilon (a^\mu) = 0

\varepsilon (_\nu) =_\nu \,

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

抄文引用元・出典: フリー百科事典『ウィキペディア（Wikipedia）』
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