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K-Poincaré algebra

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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 =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 =0,\;=i\varepsilon _{jkl}P_{l},\;=i\varepsilon _{jkl}N_{l},\;=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 =iP_{j},\;=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},\;=-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

  1. Majid, S.; Ruegg, H. (1994). "Bicrossproduct structure of κ-Poincare group and non-commutative geometry". Physics Letters B. 334 (3–4). Elsevier BV: 348–354. arXiv:hep-th/9405107. Bibcode:1994PhLB..334..348M. doi:10.1016/0370-2693(94)90699-8. ISSN 0370-2693.


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