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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
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.
References