III.75 Quantum Groups

  Shahn Majid

There are at least three different paths that lead to the objects known today as quantum groups. They could be summarized briefly as quantum geometry, quantum symmetry, and self-duality. Any one of them would be a great reason to invent quantum groups and each of them had a role in the development of the modern theory.

1 Quantum Geometry

One of the great discoveries in physics in the last century was that classical mechanics should be replaced by quantum mechanics, in which the space of possible positions and momenta of a particle is replaced by the formulation of position and momentum as mutually noncommuting operators. This noncommutativity underlies Heisenberg’s “uncertainty principle,” but it also suggests the need for a more general notion of geometry in which coordinates need not commute. One approach to noncommutative geometry is discussed in OPERATOR ALGEBRAS [IV.15 §5]. However, another approach is to note that geometry really grew out of examples such as spheres, tori, and so forth, which are LIE GROUPS [III.48 §1] or objects closely related to Lie groups. If one wants to “quantize” geometry, one should first think about how to generalize basic examples like this: in other words, one should try to define “quantum Lie groups” and associated “quantum” homogeneous spaces.

The first step is to consider geometrical structures not so much in terms of their points but in terms of corresponding algebras. For example, the group SL₂ () is defined as the set of 2 × 2 matrices of complex numbers such that αδ—βγ = 1. We can think of this as a subset of ⁴, and indeed not just a subset but a VARIETY [III.95]. The natural class of functions associated with this variety is the set of polynomials in four variables (which are defined on ⁴) restricted to the variety. However, if two polynomials take equal values on the variety, then we identify them. In other words, we take the algebra of polynomials in four variables a, b, c, and d and QUOTIENT [I.3 §3.3] by the IDEAL [III.81 §2] generated by the polynomial ad – bc – 1. (This construction is discussed in detail in ARITHMETIC GEOMETRY [IV.5 §3.2].) Let us call the resulting algebra [SL2].

We can do the same for any subset X ⊂ ⁿ that is defined by polynomial relations. This gives us a precise one-to-one correspondence between subsets of this type and certain commutative algebras equipped with n generators. Let us write [X] for the algebra that corresponds to X. As with many similar constructions (see, for example, the discussion of adjoint maps in DUALITY [III.19]), a suitable map from X to Y gives rise to a map from [Y] to [x]. More precisely, the map ϕ from X to Y has to be polynomial (in a suitable sense) and the resulting map from [Y] to [X] is an algebra homomorphism ϕ^* that satisfies the formula ϕ^*(p) (x) = p(ϕx) for every x ∈ X and p ∈ [Y].

Going back to our example, the set SL₂() has a group structure SL₂() × SL₂() → SL₂() defined by the matrix product. The set SL₂() × SL₂() is a variety in ⁸ and the matrix product depends in a polynomial way on the entries in the matrices, so we obtain an algebra homomorphism Δ : [SL₂] → [SL₂] [SL₂], which is known as the coproduct. (The algebra [SL₂] [SL₂] is isomorphic to [SL₂ × SL₂].) It turns out that Δ can be expressed by the formula

This formula needs a word or two of explanation: the variables a, b, c, and d are the four generators of the algebra of polynomials in four variables (and hence of its quotient by ad – bc – 1), and the right-hand side is a shorthand way of saying that Δa = a a + b c, and so on. Thus, Δ is defined on the generators by a sort of mixture of TENSOR PRODUCTS [III.89] and matrix multiplication.

One can then show that the associativity of matrix multiplication in SL₂ is equivalent to the assertion that (Δ id)Δ = (id Δ)Δ. To understand what these expressions mean, bear in mind that Δ takes elements of [SL₂] to elements of [SL₂] [SL₂]. Thus, when we apply the map (Δ id)Δ, for example, we begin by applying Δ, and thereby creating an element of [SL₂] [SL₂]. This element will be a linear combination of elements of the form p q, each of which will then be replaced by Δp q.

Similarly, one can express the rest of the group structure of SL₂() equivalently in terms of the algebra [SL₂]. There is a counit map ε : [SL₂] → k, which corresponds to the group identity, and an antipode map S : [SL₂] → [SL₂], which corresponds to the group inversion. The group axioms appear as equivalent properties of these maps, making [SL₂] into a “Hopf algebra” or “quantum group.” The formal definition is as follows.

Definition. A Hopf algebra over a field k is a quadruple (H, Δ, , S), where

    (i) H is a unital algebra over k;

    (ii) Δ : H → H H, : H → k are algebra homomorphisms such that (Δ id)Δ = (idΔ)Δ and ( id)Δ = (id )Δ = id;

    (iii) S : H → H is a linear map such that m(id S) Δ = m(S id) Δ = l, where m is the product operation on H.

There are two great things about this formulation. The first is that the notion of a Hopf algebra makes sense over any field. The second is that nowhere did we demand that H was commutative. Of course, if H is derived from a group, then it certainly is commutative (since multiplying two polynomials is commutative), so if we can find a noncommutative Hopf algebra, then we have obtained a strict generalization of the notion of a group. The great discovery of the past two decades is that there are indeed many natural noncommutative examples.

For example, the quantum group _q[SL₂] is defined as the free associative noncommutative algebra on symbols a, b, c, and d modulo the relations

ba = qab, bc = cb, ca = qac, dc = qcd, db = qbd, da = ad + (q - q^-1)bc, ad - q^-1 bc = 1.

This forms a Hopf algebra with Δ given by the same formula as it is for [SL₂] and with suitable maps and S. Here q is a nonzero element of , and as q → 1 one obtains [SL₂]. This example generalizes to canonical examples _q[G] for all complex simple Lie groups G.

Much of group theory and Lie group theory can be generalized to quantum groups. For example, Haar integration is a linear map ∫ : H → k that is translation invariant in a certain sense that involves Δ. If it exists, it is unique up to a scalar multiple, and it does indeed exist in most cases of interest, including all finite-dimensional Hopf algebras. Likewise, the notion of a complex Of DIFFERENTIAL FORMS [III.16] (Ω, d) makes sense over any algebra H as a proxy for a differential structure. Here, Ω = _n Ωⁿ is required to be an associative algebra generated by Ω⁰ = H and Ω¹, but one does not assume that it is graded-commutative as in the classical case. When H is a Hopf algebra one can ask that Ω is translation invariant, again in a certain sense that involves the coproduct Δ. In this case both Ω and its COHOMOLOGY [IV.6 §4] as a complex are super (or graded) quantum groups. The axioms of a (graded) Hopf algebra were originally introduced by Heinz Hopf in 1947 precisely to express the structure of the cohomology ring of a group, so this result brings us back full circle to the origins of the subject. For most quantum groups, including all the _q[G], one has a natural minimal complex (Ω, d). Thus, a “quantum group” is not merely a Hopf algebra but has additional structure analogous to that of a Lie group.

There are many other quantum groups that are not related to q-deformations. There are also applications of the theory to finite groups. If G is a finite group, one has a corresponding algebra k(G) of all functions on G with pointwise product and a coproduct (Δf)(g,h) = f(gh) for f ∈ k(G) and g, h ∈ G. Here we identify k(G) k(G) and k(G × G), which makes Δ f into a function of two variables, and one may check even more simply that this is a Hopf algebra. There can never be an interesting classical differential structure on a finite set, but if we use the methods developed for quantum groups, then we have one or more translation-invariant complexes (Ω¹, d) on any finite group. Applying further parts of the theory of quantum group differential geometry, one finds, for example, that the alternating group A₄ is naturally Ricci-flat, while the symmetric group S₃ naturally has constant CURVATURE [III.13], much like a 3-sphere.

2 Quantum Symmetry

Symmetry in mathematics is usually expressed as the action of a group or Lie algebra of finite or infinitesimal transformations of some structure. If you have a collection of transformations that is closed under inversion and composition, then you necessarily have an ordinary group. So how might one generalize this? The answer is that one begins by observing that a group G can act on several objects at the same time. If a group acts on two objects X and Y, then it also acts on their direct product X × Y, with g(x,y) = (gx, gy). Here we are making implicit use of a diagonal or “duplication” map Δ : G → G × G, which duplicates a group element so that one copy can act on the first object and the other on the second object. In order to generalize this it once again pays to replace the notion of a group G by that of an algebra. This time we use the group algebra kG, which is the set of all formal linear combinations Σ_i λ_ig_i, where the g_i are elements of G and the λ_i are scalars from the field k. The elements of G (considered as particularly simple linear combinations of this kind) form a basis of kG and we multiply them as we would in G itself. One then extends this definition to products of more general linear combinations in the obvious way. We also extend Δ linearly from Δg = g g on the basis elements to a map from kG to kG kG. Together with some associated maps and S, this makes kG into a Hopf algebra. Note that this is a completely different use of the coproduct from the one in the previous section, since the group product has already gone into the algebra. One has a similar story for the “enveloping algebra” U(g) associated with any Lie algebra g; this is generated by a basis of g with certain relations and becomes a Hopf algebra with the coproduct Δξ = ξ 1 + 1 ξ “sharing out” an element ξ ∈ g for the purposes of acting on a tensor product of objects on which g acts.

Extrapolating from these two examples, a general “quantum symmetry” means an algebra H equipped with further structure Δ that allows one to form a tensor product V W of any two representations V, W of the algebra in an associative manner. An element h ∈ H acts as h(v w) = (Δh) (v w), where one part of Δh acts on v ∈ V and another part on w ∈ W. This is a second route to the Hopf algebra axioms we gave in the previous section.

Note that, in the examples just given, Δ has had a symmetric output. As a consequence, if V and W are representations of a group or Lie algebra, then V W and W V are isomorphic via the obvious map that takes v w to w v. In general, however, V W and W V may be unrelated, so it is now the tensor product that is being made noncommutative. In nice examples it may be the case that V W ≅ W V, but not necessarily by the obvious map. Instead, there may be a nontrivial isomorphism for every pair V, W, which may nevertheless obey some reasonable conditions. This happens for a large class of examples, denoted by U_q(g) and associated with all complex simple Lie algebras. For these examples, the isomorphism obeys the braid or Yang–Baxter relations among any three representations (see BRAID GROUPS [III.4]). As a result, these quantum groups lead to KNOT AND 3-MANIFOLD INVARIANTS [III.44] (the Jones knot invariant comes from the example U_q(sl₂), where sl₂ is the Lie algebra of the group SL₂()). The parameter q can usefully be regarded here as a formal variable, and these examples can be thought of as some kind of deformation of the classical enveloping algebras U_(g). They arose originally in work of Drinfeld and of Jimbo in the theory of quantum integrable systems.

3 Self-duality

A third point of view is that Hopf algebras are the next simplest CATEGORY [III.8] after Abelian groups of structures that admit a FOURIER TRANSFORM [III.27]. It is not immediately obvious, but the axioms (i)–(iii) in the definition we gave earlier have a certain symmetry. One can write out the requirement (i) of a unital algebra H in terms of linear maps m : H H → k and η : k → H (here η specifies the identity element of H as the image of 1 ∈ k) that have to obey some straightforward commutative diagrams. If you reverse all the arrows in these diagrams, then you have the axioms displayed in (ii), obtaining what could be called a “coalgebrs.” The requirement that the coalgebra structures Δ and are algebra maps is given by a collection of diagrams that is invariant under arrow reversal. Finally, the axioms in (iii), as commutative diagrams, are invariant under arrow reversal in the above sense.

Thus, the axioms of a Hopf algebra have the special property of being symmetric under arrow reversal. A practical consequence is that if H is a finite-dimensional Hopf algebra, then so is H*, with all structure maps defined as the adjoints of those of H (which necessarily reverses arrows). In the infinite-dimensional case one needs a suitable topological dual, or one can just speak of two Hopf algebras as dually paired to each other. For instance, _q[SL₂] and U_q(sl₂) above are dually paired, while if G is finite then (kG)* = k(G), the Hopf algebra of functions on G.

As an application, let H be finite dimensional with basis {e_a}, let H* have a dual basis {f^a}, and let ∫ denote a right-translation-invariant integral on H. The Fourier transform : H → H* is defined as

and has many remarkable properties. A special case is a Fourier transform : k(G) → kG for any finite group G, which does not have to be Abelian. If G happens to be Abelian, then kG ≅ k(), where is the group of characters, and we recover the usual Fourier transform for finite Abelian groups. The point is that in the non-Abelian case, kG is not commutative and hence not the algebra of functions on any usual “Fourier dual” space.

Figure 1 Putting quantum groups in context. Self-dual categories are shown on the horizontal axis.

This point of view is responsible for the second main class of genuine quantum groups to have been discovered, namely the “bicrossproduct” ones of self-dual form. They are simultaneously “coordinate” and “symmetry” algebras, and are truly connected with quantum mechanics. An example, which is written

is the so-called Poincaré quantum group of a certain noncommutative spacetime with coordinates x, y, z, t, where t does not commute with the other variables. This quantum group can also be interpreted as the quantization of a particle moving in a curved geometry with black-hole-like features. In essence, the self-duality of quantum groups provides a paradigm for “toy models” of the unification of gravity (as spacetime geometry) and quantum theory.

This is part of a wider picture indicated in figure 1. A category of objects with a coherent notion of “tensor product” is called a monoidal (or tensor) category, and we have seen that this is the case for representations of quantum groups. There, one also has a “forgetful functor” to the category of vector spaces, which forgets the quantum group action. This embeds quantum groups into the next most general self-dual category (in a representation-theoretic sense), namely that of functors between monoidal categories. Over on the right, I have included Boolean algebras as primitive structures with (de Morgan) duality. However, the connection between duality here and the other dualities is speculative.

Further Reading

Majid, S. 2002. A Quantum Groups Primer. London Mathematical Society Lecture Notes, volume 292. Cambridge: Cambridge University Press.