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In Preskill's note, 9.1.2 in page 44, concerning the fusion space, it states that:

The fusion rules of the model specify the possible values of the total charge $c$ when the constituents have charges $a$ and $b$. These can be written $$a \times b = \sum_c N^c_{ab} c $$ where each $N^c_{ab}$ is a nonnegative integer and the sum is over the complete set of labels. Note that $a$, $b$ and $c$ are labels, NOT vector spaces; the product on the left-hand side is NOT a tensor product and the sum on the right-hand side is NOT a direct sum. Rather, the fusion rules can be regarded as an abstract relation on the label set that maps the ordered triple $(a, b; c)$ to $N^c_{ab} c$.

See after (9.66), Preskill stress again: We emphasize again, however, that while the fusion rules for group representations can be interpreted as a decomposition of a tensor product of vector spaces as a direct sum of vector spaces, in general the fusion rules in an anyon model have no such interpretation.

However, people often write the fusion rule as $$a \otimes b = \oplus_c N^c_{ab} c$$ with the tensor product $\otimes$ and the direct sum $\oplus$.

I am gathering people's comment: Is this just a matter of taste of notations? Or are these $\times,\otimes$, or $+,\oplus$ really implying different physical meaning? Which one is correct?

See also this post: direct-sum-of-anyons, there they use the direct sum.

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Mathematicians like to write tensor product, since in many cases (or maybe in all cases) anyon types (simple objects) are indeed irreducible representations of some algebraic object (e.g. Hopf algebra, quantum groups), and irreducible representations of finite groups provide a large family of examples for fusion categories, where $\otimes$ and $\oplus$ really mean tensor product and direct sum. Of course in general things are much more abstract, but the notations still remain.

Physicists are usually a little sloppy about the notations. It is probably a personal choice to write $\times$ or $\otimes$.

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  • $\begingroup$ Every fusion category is monoidally equivalent to the representation category of some weak Hopf algebra. This is in one of the early Ostrik papers. Module Categories, Weak Hopf Algebras, and Modular Invariants I think. $\endgroup$ – Matthew Titsworth Apr 25 '15 at 2:21
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I was also confused about this point. And my current understanding is, these are two different notion, let me start with the second one, inside a fusion category, it is better to write in the following way:

$$a\otimes b = \bigoplus_c V^c_{ab} c$$ where a,b,c are objects, and $V^c_{ab}$ is the vector space representing the morphisms from a tensor b to c. (It is a vector space because of the definition of fusion category require the the morphisms have a linear structure, or shortly speaking, it is a linear category) While the non-negative integer $N_{ab}^c=dim(V^c_{ab})$ is the dimension of the vector space (physically speaking, it might be called fusion or splitting space, c.f. Kitaev). So the $\otimes$ and $\oplus$ used here is suitable and meaningful, you can consider this notation as a generalization of usual tensor product and direct sum in linear algebra, because the objects $a,b,c$ could be understood as a generalization of usual vector spaces.

Now come to the fusion algebra you wrote $$a\times b= \sum_c N^c_{ab} c$$ It is a little confusing using the same notation, but I guess the precise meaning of this is the "Grothendieck ring" of the previous fusion category:$$\mathcal{C}\rightarrow \mathcal{K}(\mathcal{C})$$ You can treat this as a way to extract some "invariants" from the fusion category, by sending objects to equivalent classes, and take the dimension of the vector spaces $N^c_{ab}=dim(V^c_{ab})$. By this procedure, you might loss some information, $i.e.$ you can possibly have two different fusion category $\mathcal{C}$ and $\mathcal{C}'$ having the same Grothendieck ring $\mathcal{K}(\mathcal{C})=\mathcal{K}(\mathcal{C}')$. Or equivalently speaking, you might have different "categorification" of a single ring(fusion algebra here). The use of terminology "categorification" is not rigorous here, but the idea is simply replacing some low level data, like integer $N$ by a higher level data, like vector space $V$, with the dimension $N$.

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