  We consider the exact S-matrix governing the planar spectral problem for
strings on $AdS_5\times S^5$ and $\mathcal N=4$ super Yang-Mills, and we show
that it is invariant under a novel "boost" symmetry, which acts as a
differentiation with respect to the particle momentum. This generator leads us
also to reinterpret the usual centrally extended $\mathfrak{psu}(2|2)$
symmetry, and to conclude that the S-matrix is invariant under a $q$-Poincar\'e
supersymmetry algebra, where the deformation parameter is related to the 't
Hooft coupling. We determine the two-particle action (coproduct) that turns out
to be non-local, and study the property of the new symmetry under crossing
transformations. We look at both the strong-coupling (large tension in the
string theory) and weak-coupling (spin-chain description of the gauge theory)
limits; in the former regime we calculate the cobracket utilising the universal
classical r-matrix of Beisert and Spill. In the eventuality that the boost has
higher partners, we also construct a quantum affine version of 2D Poincar\'e
symmetry, by contraction of the quantum affine algebra
$U_q(\widehat{\mathfrak{sl}_2})$ in Drinfeld's second realisation.
