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**A note on free composition of subfactors.**
*(English)*
Zbl 0968.46045

Andersen, Jørgen Ellegaard (ed.) et al., Geometry and physics. Proceedings of the conference at Aarhus University, Aarhus, Denmark, 1995. New York, NY: Marcel Dekker. Lect. Notes Pure Appl. Math. 184, 339-361 (1997).

In an unpublished paper the authors have introduced the associative algebra \(FC_n(a,b)\) which is the linear span of the set of diagrams that arise when the points of two identical sequences of colored points located on two parallel lines are connected. There are two colors which are distinguished by two complex numbers \(a\) and \(b\). Multiplication of two diagrams is defined by concatenation (similar to braids). The first part resumes the basic results about \(FC_n(a,b)\) and its irreducible representations in the semisimple case. Next they give an explicit formula for the dimension of irreducible representations. In the second part of the paper the authors explain the connection with the theory of subfactors: The algebras \(FC_n(a,b)\) arise as higher relative commutants of subfactors that are “freely composed” to give a subfactor with an associated fusion algebra and help to clarify the structure of such a fusion algebra.

For the entire collection see [Zbl 0855.00020].

For the entire collection see [Zbl 0855.00020].

Reviewer: H.Schröder (Dortmund)

### MSC:

46L37 | Subfactors and their classification |