This chapter is devoted to the infinitesimal or semiclassical structures underlying the theory of quantum groups. The infinitesimal notion of a group is that of a Lie algebra: we follow the same line now for a Hopf algebra or quantum group. Another point of view which we consider comes from classical and quantum mechanics: we can think of a Hopf algebra as a noncommutative deformation of the commutative algebra of functions ℂ(G) on a group G, recovered as the limit t → 0, say, of a deformation parameter t. In this case one can imagine an order-by-order expansion in powers of t. Since t controls the degree of noncommutativity, it plays a role mathematically analogous to that of Planck's constant ħ in some approaches to quantisation. Recall that, in quantum mechanics, the algebra of observables is a noncommutative version of the classical algebra of functions on phase-space. The data governing the lowest order of deformation of this are contained in the semiclassical theory. They consists in our case of geometrical structures such as Poisson brackets on the group G.

In fact, we have already seen Hopf algebras in Chapter 6 (the bicross product models) which really are quantisations, while for other more well known cases, such as the quantum function algebras of Chapter 4, we follow Drinfeld and take this view more as a mathematical analogy. Nevertheless, it is a very powerful analogy, and it leads to a rich theory of Poisson brackets and associated data on group manifolds.