  In this work we study the deformations into Lie bialgebras of the three
relativistic Lie algebras: de Sitter, Anti-de Sitter and Poincar\'e, which
describe the symmetries of the three maximally symmetric spacetimes. These
algebras represent the centrepiece of the kinematics of special relativity (and
its analogue in (Anti-)de Sitter spacetime), and provide the simplest framework
to build physical models in which inertial observers are equivalent. Such a
property can be expected to be preserved by Quantum Gravity, a theory which
should build a length/energy scale into the microscopic structure of spacetime.
Quantum groups, and their infinitesimal version `Lie bialgebras', allow to
encode such a scale into a noncommutativity of the algebra of functions over
the group (and over spacetime, when the group acts on a homogeneous space). In
2+1 dimensions we have evidence that the vacuum state of Quantum Gravity is one
such `noncommutative spacetime' whose symmetries are described by a Lie
bialgebra. It is then of great interest to study the possible Lie bialgebra
deformations of the relativistic Lie algebras. In this paper, we develop a
classification of such deformations in 2, 3 and 4 spacetime dimensions, based
on physical requirements based on dimensional analysis, on various degrees of
`manifest isotropy' (which implies that certain symmetries, i.e. Lorentz
transformations or rotations, are `more classical'), and on discrete symmetries
like P and T. On top of a series of new results in 3 and 4 dimensions, we find
a no-go theorem for the Lie bialgebras in 4 dimensions, which singles out the
well-known `$\kappa$-deformation' as the only one that depends on the first
power of the Planck length, or, alternatively, that possesses `manifest'
spatial isotropy.
