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.

This post imported from StackExchange Physics at 2015-04-25 19:27 (UTC), posted by SE-user Idear