In physics and mathematics, the κ-Poincaré algebra, named after Henri Poincaré, is a deformation of the Poincaré algebra into a Hopf algebra. In the bicrossproduct basis, introduced by Majid-Ruegg its commutation rules reads:
[ P μ , P ν ] = 0 {\displaystyle [P_{\mu },P_{\nu }]=0}
[ R j , P 0 ] = 0 , [ R j , P k ] = i ε j k l P l , [ R j , N k ] = i ε j k l N l , [ R j , R k ] = i ε j k l R l {\displaystyle [R_{j},P_{0}]=0,\;[R_{j},P_{k}]=i\varepsilon _{jkl}P_{l},\;[R_{j},N_{k}]=i\varepsilon _{jkl}N_{l},\;[R_{j},R_{k}]=i\varepsilon _{jkl}R_{l}}
[ N j , P 0 ] = i P j , [ N j , P k ] = i δ j k ( 1 − e − 2 λ P 0 2 λ + λ 2 | P → | 2 ) − i λ P j P k , [ N j , N k ] = − i ε j k l R l {\displaystyle [N_{j},P_{0}]=iP_{j},\;[N_{j},P_{k}]=i\delta _{jk}\left({\frac {1-e^{-2\lambda P_{0}}}{2\lambda }}+{\frac {\lambda }{2}}|{\vec {P}}|^{2}\right)-i\lambda P_{j}P_{k},\;[N_{j},N_{k}]=-i\varepsilon _{jkl}R_{l}}
Where P μ {\displaystyle P_{\mu }} are the translation generators, R j {\displaystyle R_{j}} the rotations and N j {\displaystyle N_{j}} the boosts. The coproducts are:
Δ P j = P j ⊗ 1 + e − λ P 0 ⊗ P j , Δ P 0 = P 0 ⊗ 1 + 1 ⊗ P 0 {\displaystyle \Delta P_{j}=P_{j}\otimes 1+e^{-\lambda P_{0}}\otimes P_{j}~,\qquad \Delta P_{0}=P_{0}\otimes 1+1\otimes P_{0}}
Δ R j = R j ⊗ 1 + 1 ⊗ R j {\displaystyle \Delta R_{j}=R_{j}\otimes 1+1\otimes R_{j}}
Δ N k = N k ⊗ 1 + e − λ P 0 ⊗ N k + i λ ε k l m P l ⊗ R m . {\displaystyle \Delta N_{k}=N_{k}\otimes 1+e^{-\lambda P_{0}}\otimes N_{k}+i\lambda \varepsilon _{klm}P_{l}\otimes R_{m}.}
The antipodes and the counits:
S ( P 0 ) = − P 0 {\displaystyle S(P_{0})=-P_{0}}
S ( P j ) = − e λ P 0 P j {\displaystyle S(P_{j})=-e^{\lambda P_{0}}P_{j}}
S ( R j ) = − R j {\displaystyle S(R_{j})=-R_{j}}
S ( N j ) = − e λ P 0 N j + i λ ε j k l e λ P 0 P k R l {\displaystyle S(N_{j})=-e^{\lambda P_{0}}N_{j}+i\lambda \varepsilon _{jkl}e^{\lambda P_{0}}P_{k}R_{l}}
ε ( P 0 ) = 0 {\displaystyle \varepsilon (P_{0})=0}
ε ( P j ) = 0 {\displaystyle \varepsilon (P_{j})=0}
ε ( R j ) = 0 {\displaystyle \varepsilon (R_{j})=0}
ε ( N j ) = 0 {\displaystyle \varepsilon (N_{j})=0}
The κ-Poincaré algebra is the dual Hopf algebra to the κ-Poincaré group, and can be interpreted as its “infinitesimal” version.
References