Commutative Algebras in Fibonacci Categories

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.