We show how to get a non-commutative product for functions on space-time starting from the deformation of the coproduct of the Poincare group using the Drinfeld twist. Thus it is easy to see that the commutative algebra of functions on space-time (R^
4) can be identified as the set of functions on the Poincare group invariant under the right action of the Lorentz group provided we use the standard coproduct for the Poincare group. We obtain our results for the noncommutative Moyal plane by generalizing this result to the case of the twisted coproduct. This extension is not trivial and involves cohomological features. As is known, spacetime algebra fixes the coproduct on the dffeomorphism group of the manifold. We now see that the influence is reciprocal: they are strongly tied.
The Moyal and Wick-Voros planes A^{M,V}_{theta} are *-isomorphic. On each of these planes the Poincare group acts as a Hopf algebra symmetry if its coproducts are deformed by twist factors. We show that the *-isomorphism T: A^M_{theta} to A^V_{theta}
does not also map the corresponding twists of the Poincare group algebra. The quantum field theories on these planes with twisted Poincare-Hopf symmetries are thus inequivalent. We explicitly verify this result by showing that a non-trivial dependence on the non-commutative parameter is present for the Wick-Voros plane in a self-energy diagram whereas it is known to be absent on the Moyal plane (in the absence of gauge fields). Our results differ from these of (arXiv:0810.2095 [hep-th]) because of differences in the treatments of quantum field theories.
