Reductions of Tensor Categories Modulo Primes

Abstract

We study good (i.e., semisimple) reductions of semisimple rigid tensor categories modulo primes. A prime p is called good for a semisimple rigid tensor category 𝒞 if such a reduction exists (otherwise, it is called bad). It is clear that a good prime must be relatively prime to the Müger squared norm |V|² of any simple object V of 𝒞. We show, using the Ito–Michler theorem in finite group theory, that for group-theoretical fusion categories, the converse is true. While the converse is false for general fusion categories, we obtain results about good and bad primes for many known fusion categories (e.g., for Verlinde categories). We also state some questions and conjectures regarding good and bad primes.

Key Words:

• Good prime
• Reduction modulo a prime
• Tensor categories

2000 Mathematics Subject Classification:

• Primary 16W30
• Secondary 18D10

ACKNOWLEDGMENTS

We are very grateful to Noah Snyder for useful discussions, in particular for contributing Example 4.5. The research of the first author was partially supported by the NSF grant DMS-1000113. The second author was supported by The Israel Science Foundation (grant No. 317/09). Both authors were supported by BSF grant No. 2008164.

Notes

For the reader's convenience, let us give explicit expressions of A(T ₂) and A(T ₄) in terms of m and b. Let m _*: X → X ⊗ X be the map obtained from the dual of m by identifying X* with X using b. Then we have

Using that if X is the adjoint representation of 𝔰𝔩(2) then m is the Lie bracket and m _* is its dual under the Killing form, we arrive at formula (1).

As was explained to us by Noah Snyder, it can also be shown, using a specific presentation of the 6-object Haagerup category (see [12]), that any other prime is good for this category.

Communicated by S. Montgomery.

Dedicated to Miriam Cohen.