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 μ {\displaystyle a^{\mu }} and Λ μ ν {\displaystyle {\Lambda ^{\mu }}_{\nu }} with the usual constraint:
η ρ σ Λ μ ρ Λ ν σ = η μ ν , {\displaystyle \eta ^{\rho \sigma }{\Lambda ^{\mu }}_{\rho }{\Lambda ^{\nu }}_{\sigma }=\eta ^{\mu \nu }~,}
where η μ ν {\displaystyle \eta ^{\mu \nu }} is the Minkowskian metric:
η μ ν = ( − 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ) . {\displaystyle \eta ^{\mu \nu }=\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 {\displaystyle [a_{j},a_{0}]=i\lambda a_{j}~,\;[a_{j},a_{k}]=0}
[ a μ , Λ ρ σ ] = i λ { ( Λ ρ 0 − δ ρ 0 ) Λ μ σ − ( Λ α σ η α 0 + η σ 0 ) η ρ μ } {\displaystyle [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 μ {\displaystyle a^{\mu }} and Λ μ ν {\displaystyle {\Lambda ^{\mu }}_{\nu }} are particularly simple. The Lorentz generator in this case is:
Λ μ ν = ( cosh τ sinh τ sinh τ cosh τ ) {\displaystyle {\Lambda ^{\mu }}_{\nu }=\left({\begin{array}{cc}\cosh \tau &\sinh \tau \\\sinh \tau &\cosh \tau \end{array}}\right)}
and the commutation rules reads:
[ a 0 , ( cosh τ sinh τ ) ] = i λ sinh τ ( sinh τ cosh τ ) {\displaystyle [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 , ( cosh τ sinh τ ) ] = i λ ( 1 − cosh τ ) ( sinh τ cosh τ ) {\displaystyle [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:
Δ a μ = Λ μ ν ⊗ a ν + a μ ⊗ 1 {\displaystyle \Delta a^{\mu }={\Lambda ^{\mu }}_{\nu }\otimes a^{\nu }+a^{\mu }\otimes 1}
Δ Λ μ ν = Λ μ ρ ⊗ Λ ρ ν {\displaystyle \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 μ ) = − ( Λ − 1 ) μ ν a ν {\displaystyle S(a^{\mu })=-{(\Lambda ^{-1})^{\mu }}_{\nu }a^{\nu }}
S ( Λ μ ν ) = ( Λ − 1 ) μ ν = Λ ν μ {\displaystyle S({\Lambda ^{\mu }}_{\nu })={(\Lambda ^{-1})^{\mu }}_{\nu }={\Lambda _{\nu }}^{\mu }}
ε ( a μ ) = 0 {\displaystyle \varepsilon (a^{\mu })=0}
ε ( Λ μ ν ) = δ μ ν {\displaystyle \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.
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