 Dilemma: Fusion space from a direct sum of anyons or NOT

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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?

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

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The fusion algebra is an associative algebra structure on the fusion space - an abstract space with a basis consisting of labels for the irreducible representations, and authors can write the product in any way they like, as long as they are consistent. In the case of representations of semisimple groups or Lie algebras it is the representation ring. The notation is therefore usually that used for addition and multiplication of a ring.

On the other hand, on a more elementary level, the Clebsch-Gordan decompositions and branching rules for groups or Lie algebras are usually introduced just as rules for decomposing tensor products etc.; then one uses $\bigotimes$ and $\bigoplus$, and treats the labels as standing for the representation spaces (modules) themselves.

In quantum field theories, the representations are typically not closed under forming tensor products but only under forming deformed tensor products; this is meant by ''in general the fusion rules in an anyon model have no such interpretation''. The deformed tensor product is constructed according to the coproduct of an associated weak Hopf algebra; see Rehren. In a categorial description of this, the notation for the products of modules is again the tensor product symbol. See the paper Module categories, weak Hopf algebras and modular invariants by V. Ostrik mentioned in the comment by Matthew Titsworth.

answered Apr 26, 2015 by (13,969 points)

@ Arnold Neumaier thanks, +1.

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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 $\otimes$.

This post imported from StackExchange Physics at 2015-04-25 19:27 (UTC), posted by SE-user Meng Cheng
answered Apr 24, 2015 by (550 points)
@ Meng. Thanks for the comment.

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

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

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