# Fibonacci Anyons

## 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} = f_{n-1} ?* 1 ®

*f*, where the

_{n}? т*f*’s are the Fibonacci numbers defined by the recursion

*f*= 1, f

_{-1}_{0}= 0, and f

_{n}+i =

*f*+ f

_{n}_{n}-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 (У_{1}, *V _{t})* 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.