K-Poincaré group explained

In physics and mathematics, the κ-Poincaré group, named after Henri Poincaré, is a quantum group, obtained by deformation of the Poincaré group into a Hopf algebra.It is generated by the elements

a ^{\mu} and { Λ ^{\mu}} _{\nu} with the usual constraint:η ^{\rho} { Λ ^{\mu}} _{\rho} { Λ ^{\nu}} _{\sigma} = η ^{\mu} ~, where η ^{\mu} is the Minkowskian metric:η ^{\mu} = \left(\begin{array}{cccc} -1 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1 \end{array} \right) ~.

The commutation rules reads:

[a _{j} ,a _{0]} = i λ a _{j} ~, [a _{j,a} _{k]=0} [a ^{\mu} , { Λ ^{\rho}} _{\sigma} ] = i λ \left\{ \left( { Λ ^{\rho}} _{0} - {\delta ^{\rho}} _{0} \right) { Λ ^{\mu}} _{\sigma} - \left( { Λ ^{\alpha}} _{\sigma} η _{\alpha} + η _{\sigma} \right) η ^{\rho} \right\}

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

a ^{\mu} and { Λ ^{\mu}} _{\nu} are particularly simple. The Lorentz generator in this case is:{ Λ ^{\mu}} _{\nu} = \left( \begin{array}{cc} \cosh \tau & \sinh \tau \ \sinh \tau & \cosh \tau \end{array} \right)

and the commutation rules reads:

[ a _{0} , \left( \begin{array}{c} \cosh \tau \ \sinh \tau \end{array} \right) ] = i λ ~ \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 λ \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} = { Λ ^{\mu}} _{\nu} ⊗ a ^{\nu} + a ^{\mu} ⊗ 1 \Delta { Λ ^{\mu}} _{\nu} = { Λ ^{\mu}} _{\rho} ⊗ { Λ ^{\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)} = - {( Λ ^{-1} ) ^{\mu}} _{\nu} a ^{\nu} S({ Λ ^{\mu}} _{\nu)} = {( Λ ^{-1} ) ^{\mu}} _{\nu} = { Λ ^{\mu} _{\nu}} \varepsilon (a ^{\mu)} = 0 \varepsilon ({ Λ ^{\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.