Commutative Algebras in Fibonacci Categories

TL;DR

This paper proves that Fibonacci categories and their tensor powers lack non-trivial separable commutative ribbon algebras, implying certain chiral algebras are maximal, with implications for models like Yang-Lee and specific WZW models.

Contribution

It demonstrates the complete anisotropy of Fibonacci categories and their tensor powers, establishing their maximality in the context of chiral algebras.

Findings

Fibonacci categories are completely anisotropic.

Tensor powers of Fibonacci categories have no non-trivial separable commutative ribbon algebras.

Certain chiral algebras are proven to be maximal.

Abstract

By studying NIM-representations we show that the Fibonacci category and its tensor powers are completely anisotropic; that is, they do not have any non-trivial separable commutative ribbon algebras. As an application we deduce that a chiral algebra with the representation category equivalent to a product of Fibonacci categories is maximal; that is, it is not a proper subalgebra of another chiral algebra. In particular the chiral algebras of the Yang-Lee model, the WZW models of G2 and F4 at level 1, as well as their tensor powers, are maximal.