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As far as I understand, anyons correspond to fields that live in the representation space of some (unitary?) representation of the braid group. One-dimensional representations commute and give rise to abelian anyons whereas higher-dimensional representations of the braid group give rise to non-abelian ones.

So far so good, on the other hand the "abelianess" is characterized via the fusion rules. Only if at least one of the anyons in an anyonic model has multiple fusion channels, i.e. $$ a\times b = c + d$$ we speak about non-abelian anyons. Abelian theories have in that sense unique fusion channels.

What is the connection between these two characterizations? I suppose that they are equivalent but it's not obvious to me how as fusion is fundamentally different from braiding.

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The connection between fusion and braiding in this context is that the fusion space of $n$ anyons is the vector space $V$ in the braid group representation $\rho:B(n) \rightarrow GL(V)$. For the case of two abelian anyons $a,b$, there is only one fusion channel; the vector space (and hence the braid group representation) is one-dimensional. For non-abelian anyons, the fusion space is multi-dimensional, so action of the braid group will be represented as matrices on this higher-dimensional space.

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