Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 465Similarly one studies tensor representations of U(Ξ). For example vectorfields are introduced by the transformation propertyδ ξ V α = −ξ µ (∂ µ V α ) − (∂ α ξ µ )V µ , δ ξ V α = −ξ µ (∂ µ V α ) + (∂ µ ξ α )V µ .The generalization to arbitrary tensor fields is straight forward:δ ξ T µ 1···µ nν 1···ν n= −ξ µ (∂ µ T µ 1···µ nν 1···ν n) + (∂ µ ξ µ 1)T µ···µ nν 1···ν n+ · · · + (∂ µ ξ µ n)T µ 1···µν 1···ν n−(∂ ν1 ξ ν )T µ 1···µ nν···ν n− · · · − (∂ νn ξ ν )T µ 1···µ nν 1···ν .As for scalar fields, we also find that the product of two tensors transformslike a tensor. Summarizing, we have seen that scalar fields, vectorfields and tensor fields are representations of the Hopf algebra U(Ξ), theuniversal enveloping algebra of infinitesimal diffeomorphisms. The Hopfalgebra U(Ξ) acts via infinitesimal coordinate transformations δ ξ which aresubject to the relations:[δ ξ , δ η ] = δ ξ×η ε(δ ξ ) = 0, ∆δ ξ = δ ξ ⊗ 1 + 1 ⊗ δ ξ S(δ ξ ) = −δ ξ . (3.282)The transformation operator δ ξ is explicitly given by differential operatorswhich depend on the representation under consideration. In case of scalarfields this differential operator is given by -ξ µ ∂ µ .3.17.1.4 Deformed DiffeomorphismsThe above concepts can be deformed in order to establish a consistenttensor calculus on the noncommutative space–time algebra (3.267). In thiscontext it is necessary to account the full Hopf algebra structure of theuniversal enveloping algebra U(Ξ).In our setting the algebra  possesses a noncommutative product definedby[ˆx µ , ˆx ν ] = iθ µν . (3.283)We want to deform the structure maps (3.282) of the Hopf algebra U(Ξ) insuch a way that the resulting deformed Hopf algebra which we denote byU(ˆΞ) consistently acts on Â. In the language introduced in the previoussection this means that we want  to be a U(ˆΞ)−module algebra. We claimthat the following deformation of U(Ξ) does the job. Let U(ˆΞ) be generatedas algebra by elements ˆδ ξ , ξ ∈ Ξ. We leave the algebra relation undeformedand demand [Meyer (2005)][ˆδ ξ , ˆδ η ] = ˆδ ξ×η (3.284)