nLab Ocneanu Rigidity

Idea

Every fusion category has an associated fusion ring. Ocneanu rigidity states that, up to monoidal equivalence, there are finitely many fusion categories with any given fusion ring.

This can be interpreted as saying that, when one fixes a set of fusion rules, you cannot continuously deform your category between monoidal equivalence classes.

Ocneanu rigidity is exemplary among many finiteness results for fusion categories. It is useful because it can translate many questions about fusion categories into questions about fusion rings, which are much simpler algebraic structures. For instance, it is an open problem whether there are finitely many equivalence classes of fusion categories of a given rank. Ocneanu rigidity reduces this problem to asking whether there are finitely many fusion rings of a given rank.

References

A modern statement and proof can be found here:

• Azat Gainutdinov, Jonas Haferkamp, and Christoph Schweigert. “Davydov-Yetter cohomology, comonads and Ocneanu rigidity.” Advances in Mathematics 414 (2023): 108853.

A good reference is here:

• Pavel Etingof, Dmitri Nikshych, and Viktor Ostrik. “On fusion categories.” Annals of mathematics (2005): 581-642.

The original discussion of rank-finiteness can be found here:

• Viktor Ostrik. “Fusion categories of rank 2.” arXiv preprint math/0203255 (2002).