After all the abstract algebra in the preceding two chapters, it is clearly high time for some substantial examples. This is the topic of the present chapter and the following one. In this chapter we describe quantum groups (quasitriangular Hopf algebras) that are deformations of the enveloping algebras of Lie algebras. The latter have already been introduced in Example 1.5.7 and are clearly cocommutative. But by deforming the comultiplication (and perhaps also the appearance of the multiplication), we can obtain examples that are not cocommutative and not commutative (if the Lie algebra is non-Abelian). The quasitriangular structure also becomes nontrivial. These examples can therefore be called quantum groups of enveloping algebra type. In principle, this deformation tends to be a systematic one related to the twisting construction given in Chapters 2.3–2.4, but this need not concern us as far as describing the resulting structures is concerned.

Lie groups and Lie algebras have their origin as transformations or symmetries of spaces, and this is still how they are most often used. Likewise, the quantum groups of enveloping algebra type are well suited to acting on things. In general, they act on all the things that the initial Lie algebra acts on (the action is deformed along with the quantum group structure). This makes them particularly useful in certain kinds of physical contexts. As quasitriangular Hopf algebras they have many other applications too, notably to the construction of link and three-manifold invariants, as we shall see in Chapter 9.