  A quantum deformation of the conformal algebra of the Minkowskian spacetime
in $(3+1)$ dimensions is identified with a deformation of the
$(4+1)$-dimensional AdS algebra. Both Minkowskian and AdS first-order
non-commutative spaces are explicitly obtained, and the former coincides with
the well known $\kappa$-Minkowski space. Next, by working in the conformal
basis, a new non-commutative Minkowskian spacetime is constructed through the
full (all orders) dual quantum group spanned by deformed Poincar\'e and
dilation symmetries. Although Lorentz invariance is lost, the resulting
non-commutative spacetime is quantum group covariant, preserves space isotropy
and, furthermore, can be interpreted as a generalization of the
$\kappa$-Minkowski space in which a variable fundamental scale (Planck length)
appears.

High Energy Physics - Theory;General Relativity and Quantum Cosmology;Quantum Algebra