Quantization of inhomogeneous Lie bialgebras

A self-dual class of Lie bialgebra structures (g, g*) on inhomogeneous Lie algebras g describing kinematical symmetries is considered. In that class, both g and g* split into the semi-direct sums g = h (sic) and g* = h* (sic)* with abelian ideals of translations v and h*. We build the explicit quantization of the universal enveloping algebra U (g), including the coproduct, commutation relations among generators, and, in case of coboundary g, the universal R-matrix. This class of Lie bialgebras forms a self-dual category stable under the classical double procedure. The quantization turns out to be a functor to the category of Hopf algebras which commutes with operations of dualization and double. (C) 2002 Elsevier Science B.V. All rights reserved.