Infinite Braided Tensor Products and 2d Quantum Gravity

Braided tensor products have been introduced by the author as a systematic way of making two quantum-group-covariant systems interact in a covariant way, and used in the theory of braided groups. Here we study infinite braided tensor products of the quantum plane (or other constant Zamolodchikov algebra). It turns out that such a structure precisely describes the exchange algebra in 2D quantum gravity in the approach of Gervais. We also consider infinite braided tensor products of quantum groups and braided groups. 1. Of central importance in the theory of braided groups initiated in [6][7][8][9][10] is the notion of covariant algebras and their braided tensor products. Here we give a new application of this notion. Let us recall that a covariant algebra is simply an algebra B on which a quantum group H acts (or a dual quantum group A of function algebra type, coacts) in such a way that the product and unit of B are covariant, i.e. the maps · : B⊗B → B and η : C → B respectively are interwiners for the quantum-group (co)action. This is obviously an important notion if we want to work with physical systems (described by algebras) with quantum-group symmetry. A fancy way to think of covariance is that the algebra B lives in the category of (co)-modules of the quantum group[7, Sec. 6]: ‘Working in the category’ just says that we keep everything manifestly covariant. Thinking about things this way allows us to treat covariance in the same way as we treat super-symmetry (we make everything Z2-graded). This unification between usual notions of group covariance and super-symmetry is one of the remarkable unifications made possible by quantum groups[7, Sec. 6]. In particular, the (co)modules of a true quantum group (with universal R-matrix or its dual concept for a quantum function algebra) have a braiding Ψ[5, Sec. 7]. This is a coherent collection of maps ΨV,W : V ⊗W → W ⊗V allowing the ‘transposition’ of any two objects in the category (any two representations) with properties like the usual transposition or super-transposition ΨV,W (v⊗w) = (−1) |v||w|w⊗ v, except that typically we no longer have ΨV,W = Ψ −1 W,V (so they are best represented by braids rather than by permutations). The main lemma which will concern us, which is the fundamental lemma of the theory of braided groups is: if B,C are covariant algebras then there is a braided tensor product B⊗C which is again a covariant algebra. Thus, we have a way of combining covariant systems in a covariant way. This braided tensor product algebra is B⊗C as a space, but product (b⊗ c) · (d⊗ e) = bΨC,B(c⊗ d)e, b, d ∈ B, c, e ∈ C (1) where we mean to first apply ΨC,B to c⊗ d ∈ C⊗B and multiply the result from the left in B and from the right in C. That this is associative follows from functoriality and hexagon identities for Ψ. See [8] for a diagrammatic proof. The braided tensor product B⊗C contains B and C as subalgebras, but unlike the usual tensor product, these subalgebras do not commute. Instead, their exchange is given by the braiding Ψ. This braided tensor product is like the super-tensor product of super-algebras, leading us to interpret (1) as saying that the elements in the algebras B,C have braid statistics SERC Fellow and Drapers Fellow of Pembroke College, Cambridge