K Poincaré group - Alchetron, The Free Social Encyclopedia

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 and Λ μ ν with the usual constraint:

η ρ σ Λ μ ρ Λ ν σ = η μ ν ,

where η μ ν is the Minkowskian metric:

η μ ν = ( − 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ) .

The commutation rules reads:

[ a j , a 0 ] = i λ 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 }

In the (1 + 1)-dimensional case the commutation rules between a μ and Λ μ ν are particularly simple. The Lorentz generator in this case is:

Λ μ ν = ( cosh ⁡ τ sinh ⁡ τ sinh ⁡ τ cosh ⁡ τ )

and the commutation rules reads:

[ a 0 , ( cosh ⁡ τ sinh ⁡ τ ) ] = i λ sinh ⁡ τ ( sinh ⁡ τ cosh ⁡ τ )

[ a 1 , ( cosh ⁡ τ sinh ⁡ τ ) ] = i λ ( 1 − cosh ⁡ τ ) ( sinh ⁡ τ cosh ⁡ τ )

The coproducts are classical, and encode the group composition law:

Δ a μ = Λ μ ν ⊗ a ν + a μ ⊗ 1

Δ Λ μ ν = Λ μ ρ ⊗ Λ ρ ν

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

S ( Λ μ ν ) = ( Λ − 1 ) μ ν = Λ ν μ

ε ( a μ ) = 0

ε ( Λ μ ν ) = δ μ ν

The κ-Poincaré group is the dual Hopf algebra to the K-Poincaré algebra, and can be interpreted as its “finite” version.

References

K-Poincaré group Wikipedia

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