Unitary ribbon data

A unitary ribbon fusion category is a unitary fusion category with a unitary braiding and a ribbon structure in the sense of Twist and ribbon structure. Thus the twist $\theta$ is a natural automorphism of the identity satisfying the balancing identity and the dual-compatibility condition of that root definition, and each component is unitary: ${\theta }_X^{†}={\theta }_X^{-1}$. A ribbon structure is additional data; it is not supplied by fusion rules or by a braid-group representation alone.

The twist is unitary under these hypotheses. Let $s$ be simple, choose its left dual $s^{\vee }$, and let $\eta :\mathbf{1}\to s\otimes s^{\vee }$ be the coevaluation. A nonzero endomorphism of $s$ has zero kernel and image $s$, so it is an isomorphism and $\mathrm{End}(s)$ is a division algebra. Local finiteness makes it finite-dimensional over $\mathbb{C}$. For any $a\in \mathrm{End}(s)$, finite-dimensionality gives a nonzero polynomial $p$ with $p(a)=0$. Factor $p$ over $\mathbb{C}$; since a division algebra has no zero divisors, one linear factor must vanish, so $\mathrm{End}(s)=\mathbb{C}{\mathrm{id}}_s$. Write ${\theta }_s=\lambda {\mathrm{id}}_s$. The ribbon duality condition and linearity of dualization give ${\theta }_{s^{\vee }}=\lambda {\mathrm{id}}_{s^{\vee }}$. Naturality, ${\theta }_{\mathbf{1}}={\mathrm{id}}_{\mathbf{1}}$, and balancing give $\eta ={\theta }_{s\otimes s^{\vee }}\circ \eta ={\lambda }^2{c}_{s^{\vee },s}\circ c_{s,s^{\vee }}\circ \eta \mathrm{.}$ The double braiding is unitary, so it preserves the norm of $\eta$. The zig-zag identity gives $\eta \mathrm{\ne }0$; hence the equality implies $\mathrm{\parallel }\eta \mathrm{\parallel }=\mathrm{\mid }\lambda {\mathrm{\mid }}^2\mathrm{\parallel }\eta \mathrm{\parallel }$ and $\mathrm{\mid }\lambda \mathrm{\mid }=1$. Thus ${\theta }_s$ is unitary for every simple $s$. For any object, decompose it as a finite orthogonal sum of simples; naturality makes its twist block diagonal with these unitary components, so ${\theta }_X^{†}{\theta }_X={\mathrm{id}}_X$.