for the case of arbitrary $\lambda$.

Concluding remarks and discussion
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In the following we clarify what from our perspective is a major physical motivation for studying the universal $R$-matrix and also try to justify a type of deformation carried out in the paper which keeps the track with the Hopf algebra formalism, resulting in a mathematical structure that can be shown to be related to a Hopf algebroid [@lu].

Basically, the idea for applying a Hopf algebra formalism is grounded in the fact that the general relativity theory together with the uncertainty principle of quantum mechanics leads to a class of models with spacetime noncommutativity. In this setting the smooth spacetime geometry of classical general relativity is replaced with a Hopf algebra at the Planck scale. In our view, a particularly interesting example of Hopf algebra is $\kappa$-Poincaré Hopf algebra, which provides an algebraic setting for describing symmetry underlying the effective NCFT that results from coupling quantum gravity to matter fields after topological degrees of freedom of gravity are integrated out [@AmelinoCamelia:2003xp; @Freidel:2003sp; @Freidel:2005bb; @Freidel:2005me]. This particular type of Hopf algebra also appears in the context of Doubly Special Relativity theories [@AmelinoCamelia:2000mn; @AmelinoCamelia:2000ge; @Magueijo:2001cr], which give one possible explanation 