After the technical work of the preceding three chapters, we now take a break and return to more conceptual issues, i.e. to the general theory of Hopf algebras and quasitriangular structures. We have already seen early on, in Exercise 1.6.8, that if a Hopf algebra is represented in vector spaces V, W then it is also naturally represented in V ⊗ W. So the representations of the Hopf algebra have among themselves a tensor product operation ⊗. This is a key property of group representations, and is just as important for Hopf algebras. We have already used it several times in previous chapters in the course of our Hopf algebra constructions. In this chapter, we want to study this phenomenon quite systematically and more conceptually using the language of category theory. It should be possible to come to this chapter directly after Chapters 1 and 2, viewing the intermediate ones as providing examples and applications. For readers who want to skim this chapter, the key calculation, which also contains the essence of the entire chapter, is Theorem 9.2.4.

Category theory itself is often considered to be a hard subject by physicists, and for this reason we will try to be informal and nontechnical as much as possible. In truth, a category C just means a collection of objects (in our case they will be representations), and a specification of what it is to be an allowed map or morphism between any two objects.

In this chapter, we describe another class of examples of quantum groups, quite different from the deformations of enveloping algebras in the preceding chapter. These are the matrix quantum groups, which generalise the familiar idea of defining groups as groups of matrices. They are dual to the deformed enveloping algebra quantum groups of the Chapter 3 (we will see that they are dual quasitriangular in the sense of Chapter 2.2), and as such they are dual quantum groups in the strict sense, or quantum groups of function algebra type. In other words, rather than working with the elements of the group or Lie algebra, we work with the functions on the group (see Example 1.5.2) as our classical model. When we deform a Hopf algebra of this type so that it is no longer commutative, we can nevertheless continue to think of its elements as if they were the functions on a group in this way, even though an actual underlying group no longer exists.

This point of view is a well established one in mathematics, and comes under the general heading of noncommutative geometry. We will come to this in later chapters, but, for now, we can explain the general idea at a superficial level. It is only this superficial level that is at all relevant to the present chapter, but it can nevertheless be used as a source of motivation.

This general idea of noncommutative geometry arose in the early half of this century with an important theorem of I. Gelfand and M.A. Naimark about commutative C* algebras and the related strides in algebraic geometry that were made at the time.

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.

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.

This is the first of two physically-oriented chapters in which we take a break from R-matrices and such topics and explore instead a completely different setting in which Hopf algebras arise naturally. This is the setting of logic, quantum mechanics and probability theory, in all of which contexts the coproduct Δ of the Hopf algebra plays the role of ‘sharing out’ possibilities. The theory has potential applications in computer science as well. In Boolean algebra and quantum mechanics, the product in the algebra corresponds in some sense to deduction of facts (‘putting two and two together’), and supplementing this structure with a coproduct provides a kind of reverse of this process by sharing out possible explanations of a fact. This provides the general theme of the present chapter. The chapter can also be viewed as background material for the next chapter, where we present a number of concrete models of quantum-mechanical Hopf algebras of observables.

The simplest example of a combinatorial Hopf algebra is the shuffle algebra associated to an alphabet of symbols. Here the coproduct of a word is the formal sum of all pairs of words which could be shuffled to give the original word. Here, a shuffle of two words takes symbols from each of the words and interleaves them randomly while preserving the order of each word (this is exactly what you do when shuffling a pack of cards).

The theme of this chapter is that when one has an axiom or condition for an algebraic structure, one can consider relaxing it so that it holds only up to some ‘cocycle’ element, which is then required to obey some consistency conditions which we have to specify. The most important application of this principle will be to the condition of cocommutativity studied in Chapter 1.5. Thus, we now study a class of Hopf algebras that are cocommutative only up to conjugation by an element R, called the ‘quasitriangular structure’, and obeying some conditions. This simple idea has far-reaching implications which will recur throughout the book. From an algebraic point of view, such Hopf algebras are truly different from the Hopf algebras associated to groups or Lie algebras already encountered in Chapter 1, and yet are so close to being cocommutative that all the familiar results for groups and Lie algebras tend to have analogues here also. Hopf algebras that are almost cocommutative in this way are called quasitriangular Hopf algebras. Because their properties are so close to those of groups or Lie algebras, they are also commonly called quantum groups. We will often use this term more loosely to apply to Hopf algebras in general, but this is the strict usage. We will give examples that are indeed deformations of familiar groups and Lie algebras later, in Chapters 3 and 4, but it should not be thought that they are the only examples.

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.

Here we collect the definitions and basic constructions that will be needed throughout the book. This material is mostly quite standard for Hopf algebraists, but developed now with simplified or streamlined new proofs. The last section contains newer material. The content of the various definitions will become apparent after numerous examples and exercises. Once the unfamiliar notation is mastered, working with Hopf algebras is no harder than working with algebras alone, and is more fun. More practice will be provided when we turn to the advanced theory in Chapter 2. I would recommend that any reader should work through the present elementary chapter and at least the first part of Chapter 2 in detail since these sections are central to much of the later development. The main exception is Chapter 4 on matrix quantum groups, where one can get quite a long way purely by analogy with ordinary matrices, and perhaps Chapter 3 by analogy with Lie algebras. Thus, a viable alternative is to begin with Chapters 3 or 4 and use the present chapter and its sequel as reference or for clarification.

Algebras

This section is included to fix our notation and terminology. Any serious attempt to apply the extensive literature on Hopf algebras to physical situations will require familiarity with the basic ideas given here.

Recall that a group (G, ·) is a set G on which an associative product is defined, with a unit element e such that e · u = u = u · e for all u ∈ G, and such that every element u has an inverse, denoted u−1.

This is an introduction to the foundations of quantum group theory. Quantum groups or Hopf algebras are an exciting new generalisation of ordinary groups. They have a rich mathematical structure and numerous roles in situations where ordinary groups are not adequate. The goal in this volume is to set out this mathematical structure by developing the basic properties of quantum groups as objects in their own right; what quantum groups are conceptually and how to work with them. We will also give some idea of the meaning of quantum groups for physics. On the other hand, just as ordinary groups have all sorts of applications in physics, not one specific application but many, in the same way one finds that quantum groups have a wide variety of probably unrelated applications. This diversity is one of the themes in the volume and is a good reason to focus on quantum groups as mathematical objects.

This book is not a survey; many of the most interesting recent results in representation theory, applications in conformal field theory and lowdimensional topology, etc., are not discussed in any detail. In this sense, there is less material here than in my lecture notes [1]. In place of this fashionable material, I have developed the pedagogical side of [1], giving now more details of proofs and solutions to exercises, and in general concentrating more on that part of the theory of quantum groups that can be considered as firmly established.

This chapter is a kind of epilogue, in which we show how the machinery developed in the preceding chapters, some of it quite mathematical, can be used to provide the beginning of a kind of q-deformed geometry. It turns out that the underlying structure here is not so much a quantum group as one of the exotic braided groups which we have encountered in Chapter 9.4.2. Quantum groups still play a role as the quantum symmetry of such q-deformed spaces, but the spaces themselves tend to be braided ones. Thus we will need all the machinery developed so far in this book. Nevertheless, the problem of systematically q-deforming all the geometrical (and other) structures needed in physics is a deep and important one for physics, so we shall try to give here as self-contained and elementary a treatment as possible. It should be possible to come to this chapter directly after Chapter 4, using the intermediate chapters as reference for the mathematical underpinning when required. Also, we cover here only q-deformed or braided versions of ℝn, where the theory is fairly complete. Only when this is thoroughly understood could one reasonably expect to move on to define q-manifolds, etc. The further theory of braided geometry is deferred to a sequel to this book.

We have already covered one standard point of view on q-deforming geometrical structures in Chapter 4, namely as some kind of ‘quantisation’ of an algebra of functions.