Matched pairs of lie groups and Hopf algebra bicrossproducts

Abstract

We construct a new class of examples of non-commutative and non-cocommutative Hopf algebras. There is essentially one for every Lie bialgebra, and several other examples including one for every simple real Lie algebra. The latter are obtained from a canonical solution of the Classical Yang-Baxter Equations. Physically, these Hopf algebras arise as the algebra of observables of quantum mechanics on homogeneous spacetimes.