This chapter is devoted to the description of a large class of noncommutative and noncocommutative Hopf algebras, called bicrossproduct Hopf algebras. Although less well-known than the familiar quantum groups of Chapters 3 and 4, they have the merit of arising genuinely as the quantum algebras of observables of quantum systems. They were introduced by the author in this context, as part of an algebraic approach to unifying quantum mechanics and gravity. This is the theme of the present chapter. Of course, these noncommutative and noncocommutative Hopf algebras are also interesting on purely mathematical grounds and can be fed into any of the modern quantum groups machinery.

The idea behind the construction of these bicrossproduct Hopf algebras is self-duality. Recall from Chapter 1 that the axioms of a Hopf algebra have a remarkable input–output symmetry. We have already seen some of the physical implications of this for a quantum algebra of observables in Chapter 5.4. Another more geometrical interpretation is that of putting quantum mechanics and gravity on an equal but mutually dual footing. The reason for this is that the quantum algebra of observables is a noncommutative version of the algebra of functions C(X), where X is the classical phase-space. The degree of noncommutativity is controlled by ħ, the physical Planck's constant. On the other hand, if this is a Hopf algebra, then we have the additional structure of a coproduct, which corresponds in the classical case to a group structure, as we know from Example 1.5.2. A noncocommutative coproduct corresponds, then, to a non-Abelian group structure on phase-space, but phrased in an algebraic way that makes sense for the quantum algebra of observables as well.