The Quantum Double of a (locally) Compact Group

We generalise the quantum double construction of Drinfel’d to the case of the (Hopf) algebra of suitable functions on a compact or locally compact group. We will concentrate on the ∗-algebra structure of the quantum double. If the conjugacy classes in the group are countably separated, then we classify the irreducible ∗-representations by using the connection with so–called transformation group algebras. For finite groups, we will compare our description to the result of Dijkgraaf, Pasquier and Roche. Finally we will work out the explicit examples of SU(2) and SL(2, IR) . The quantum double of a Hopf algebra (or, double quantum group) was introduced by Drinfel’d in [9]. Quantum doubles are important examples of quasitriangular (quasi) Hopf algebras, and in that sense they are well-studied, see for instance [6], [23], [18]. The existing theory of quantum doubles has beautiful applications in physics: in [7] Dijkgraaf, Pasquier and Roche show that the representation theory covers the main interesting data of particular orbifolds of Rational Conformal Field Theories. Tightly connected are the topological interactions in spontaneously broken gauge theories. In [2], [3] Bais, Van Driel and De Wild Propitius show that the non-trivial fusion and braiding properties of the excited states in broken gauge theories can be fully described by the representation theory of a quantum double which is constructed from a finite group G via a finite dimensional Hopf algebra A = C [G] (the corresponding group ring). For a detailed treatment, see [26]. From a physical as well as a mathematical point of view it is natural to ask whether the quantum double construction from a finite group can be generalized to the case of a compact, or even locally compact group. In this report we will explicitly construct the quantum double D(G) corresponding to a (locally) compact group G. It has a natural ∗-structure, and we will find a class of ∗-representations (unitary representations), and prove rigorously that they form a complete set of irreducible ∗-representations. The construction uses the representation theory of transformation group algebras, which we will discuss in detail, and its connections with the theory of induced representations of locally ISSN 0949–5932 / $2.50 C © Heldermann Verlag 102 Koornwinder and Muller compact groups via the imprimitivity theorem of Mackey [14]. For an overview of the theory of (induced) representations of locally compact groups, see for instance Chapters 9–10–11 in [12]. In fact, the same construction works more generally for classifying the irreducible ∗-representations of transformation group algebras, see Glimm [11]. To be more precise, the construction is done in the following way: For the generalization of the quantum double, we choose the algebra Cc(G × G) of continuous functions on G × G with compact support. This allows us to use the representation theory of transformation group algebras Cc(X × G), where the locally compact group G acts continuously on a locally compact space X . Under the technical (but crucial) assumption that the conjugate action of G on G is countably separated, classification of the irreducible ∗-representations of these algebras turns out to be equivalent to classifying the irreducible representations (τ, P ) of the pair (G,O), with O an orbit of G on X . Writing G/H ' O , with H a closed subgroup of G, then (G,G/H, P ) is a system of imprimitivity for the unitary representation τ of G on a Hilbert space V , and P is a projection valued measure on G/H acting on V . From Mackey’s imprimitivity theorem it follows that such representations τ are precisely the representations of G which are induced from unitary irreducible representations α of H . The classification of irreducible ∗-representations of the quantum double is a direct consequence. We will show that the case of finite G is covered by our description, and it leads to representations which are isomorphic to the ones derived in [7]. Finally we will work out the explicit examples of G = SU(2) (compact) and G = SL(2, IR) (non–compact). Their interesting applications in physics will be discussed in a forthcoming paper by the second author, where the connection with a quantization of a Chern–Simons theory in (2+1) dimensions will be described. We are also studying in detail the coalgebra structure of the continuous quantum double, and in a follow–up of this paper we will give the tensor product decomposition (‘fusion rules’) for the quantum double representations, the universal R-matrix, and its action on a tensor product state. In fact, questions about fusion rules were our original motivation for this work. However, it soon became apparent that even the definition of the quantum double of C(G), and the classification of the irreducible representations of this quantum double had to be clarified. This led us to a thorough study of quite some older literature on transformation group algebras, which turned out to be well applicable for our case of the quantum double. To our knowledge these references have not been put together in this combination before. We do not claim originality in the contents of our main results. Notably, the main Theorem 3.9 occurs in Glimm [16], in a somewhat hidden way. Our reformulation of the theorem may be more suitable for applications, for instance in physics, and makes it possible to treat concrete examples. For expository reasons we have added our own version of the proof, with emphasis on the link with the imprimitivity theorem. It gives insight into the way we have derived the characterisation and classification of the irreducible unitary representations of the quantum double. The latter is necessary for the computation of the fusion rules and for the braiding properties (R-matrix) of the model. Koornwinder and Muller 103 1. Construction Drinfel’d [9] gives the following definition of the quantum double D(A) of a Hopf algebra A. (This definition is only mathematically precise if A is a finite dimensional Hopf algebra. However, there is a way out by working with a dual pair of bialgebras, see Majid [18], p.296.) Definition 1.1. Let A be a Hopf algebra over the field C , and A0 the dual Hopf algebra to A with the opposite comultiplication. Then D(A) is the unique quasi-triangular Hopf algebra with universal R-matrix R ∈ D(A) ⊗ D(A) such that i. As a vector space, D(A) = A⊗A0 . ii. A = A⊗ 1 and A0 = 1⊗A0 are Hopf subalgebras of D(A). iii. The mapping x ⊗ ξ 7→ xξ : A ⊗ A0 → D(A) is an isomorphism of vector spaces. Here xξ is short notation for the product (x⊗ 1)(1⊗ ξ). iv. Let (ei)i∈I be a basis of A and (e)i∈I the dual basis of A0 . Then R = ∑ i∈I (ei ⊗ 1)⊗ (1⊗ e), (1) independent of the choice of the basis. Tensor products are taken over the field C . Now write