TENSOR CATEGORIES AS A MATHEMATICAL LANGUAGE DESCRIBING QUANTUM SYMMETRIES

Abstract

The mathematical aspects of the notion of tensor categories are reviewed in connection with quantum symmetries of operator algebras. Let us begin with brief historical backgrounds for sources of tensor categories: • Abstract charcaterizations of Tannaka duals (1970–) – Algebraic Geometry (Grothendieck-Saavedra Rivano, Deligne-Milne) – Superselection Sectors (Doplicher-Haag-Roberts) • Low-Dimensional Physics (Braid statistics–Braided Tensor Categories) In all these, classical and quantum symmetries appear as tensor cateogries. Let me recall now classical Tannaka duality for compact groups. 1. Tannaka Duality Let G be a compact group and R = R(G) be the category of finite-dimensional representations of G: For G-modules V and W , Hom(V,W ) = the space of intertwiners and we have the operations of taking tensor products V,W ⇝ V ⊗W and taking conjugations V ⇝ V ∗. The group G is then recovered from the category R by g = {gV }, gV : V → V is a unitary satisfying gV⊗W = gV ⊗ gW , gV ∗ = tg−1 V , V gV −−−→ V f y yf W −−−→ gW W The fact that the category R is furnished with the operation of tensor products is abstracted into the notion of tensor cateogry. 2. Tensor Categories A tensor category is a category C such that • Hom(X,Y ) is a complex vector space, • a bivariant functor X,Y ⇝ X ⊗ Y is given, • a special object I (unit object) is given and these satisfy (X ⊗ Y )⊗ Z = X ⊗ (Y ⊗ Z), I ⊗X = X = X ⊗ I. Remark • X is just a symbol and may not be a vector space. • Commutativity X ⊗ Y = Y ⊗X is not required. In most physical applications, it is natural to require the Positivity/Unitarity: We shall work with C*-categories in which the so-called *-operation Hom(X, Y ) ∋ f 7→ f ∗ ∈ Hom(Y,X) is furnished so that f ∗f ≥ 0 in certain sense. Example 1 (i) Let N be a *-algebra (more precisely a von Neumann algebra) and consider a Hilbert space X on which N acts in a bimodule fashion. Then the totality {NXN} forms a C*-tensor category: Hom(X, Y ) = {f : X → Y ; f(aξb) = af(ξ)b, a, b ∈ N, ξ ∈ X}, X ⊗ Y = N(X ⊗N Y )N . (ii) In the theory of superselection sectors, there appear C*-tensor categories of endomorphisms: Let N be as above and consider unital *-endomorphisms of N , say ρ. Then the totality {ρ ∈ End(N)} forms a C*-tensor category by Hom(ρ, σ) = {a ∈ N ; ρ(x)a = aσ(x),∀x ∈ N}, ρ⊗ σ = ρ ◦ σ (the composite endomorphism). Remark The second example is a special case of the first one as each ρ ∈ End(N) gives rise to a bimodule NL (N)ρN , where L (N) denotes the regular representation (Hilbert space) with the left action of N by left multiplication, wheareas the right action of N is the combination of right multiplication and the endomorphism ρ. More interesting examples are provided by positive energy representations of loop groups (a geometric realization of WZW models): Let G be a compact Lie group, say SU(N), and π be a positive energy (projective) representation on a Hilbert space X of level l ≥ 1. Recall that irreducible positive energy representations of level l are parametrized by Young diagrams l = (l1, . . . , lN) satisfying l1 − lN ≤ l in such a way that the representation of G = SU(N) on the lowest energy subspace X(0) is the irreducible representation associated to l. Let N = π0(C (Sleft, G)) ′′ be the von Neumann algebra in the vacuum representation π0 of level l with Sleft denoting the left semicircle. Then X is an N -N bimodule in an obvious way and the totality {NXN} turns out to form a C*-tensor category, an operator-algebraic version of WZW-models (JonesWassermann).