Definition and Additive Structure
In this section, we consider the special case of Fibonacci anyons. These are defined by specifying the category A as follows. There are just two simple objects, 1 (the vacuum, the unit for ®) and т. Each is its own dual. (Recall that Axiom 8 requires each object to have a dual; dualization is additive, so we need only specify the duals of the simple objects.) The monoidal structure is given by т ® т = 1 ® т (plus the fact that 1 is the unit, so 1 ® т = т ® 1 = т and 1 ® 1 = 1).
The terminology “Fibonacci anyon” comes from the fact, easily verified using the distributivity of ® over ®, that iteration of ® gives т®n = fn-1 ? 1 ® fn ? т, where the f’s are the Fibonacci numbers defined by the recursion f-1 = 1, f0 = 0, and fn+i = fn + fn-i. Here and below, we use the notation k ? S to mean the sum of k copies of the object S of A. (The notation makes sense for arbitrary objects S, but we shall need it only for simple S.)
As explained in Sect. 8.4, we can identify the category A with the category of pairs (У1, Vt) of finite-dimensional complex vector spaces. Explicitly, an object X is identified with the pair (Hom(1, X), HomA X)). In particular, the unit 1 in A
is identified with (C, 0), and т is identified with (0, C). This identification respects the additive structure: ® in A corresponds to componentwise direct sum of pairs of vector spaces.