In physics and mathematics, the κ-Poincaré algebra, named after Henri Poincaré, is a deformation of the Poincaré algebra into an Hopf algebra. In the bicrossproduct basis, introduced by Majid-Ruegg its commutation rules reads:
[ P μ , P ν ] = 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 [ 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 Where P μ are the translation generators, R j the rotations and 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 Δ R j = R j ⊗ 1 + 1 ⊗ R j Δ N k = N k ⊗ 1 + e − λ P 0 ⊗ N k + i λ ε k l m P l ⊗ R m . The antipodes and the counits:
S ( P 0 ) = − P 0 S ( P j ) = − e λ P 0 P j 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 ε ( P 0 ) = 0 ε ( P j ) = 0 ε ( R j ) = 0 ε ( 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.
