Modular ribbon fusion category and Hopf-link data

Let $\mathcal{C}$ be a complex finite semisimple rigid tensor category with simple unit $\mathbf{1}$ and a finite set $I$ of isomorphism classes of simple objects. Fix a braiding $c$ and a ribbon twist $\theta$. The induced pivotal map is $j_X=u_X{\theta }_X:X\to X^{\vee \vee }$, where $u$ is the Drinfeld morphism of A braided rigid category has a Drinfeld morphism. Throughout this pair we use the associated spherical pivotal structure: $j$ is part of the data, and sphericality means ${\dim }_j(X)={\dim }_j(X^{\vee })$ for every object, as in Spherical structure. Its trace is ${\mathrm{tr}}_j(f)={\mathrm{Tr}}_L(j_{X}f)$ for $f\in \mathrm{End}(X)$.

Write $0$ for the unit label and $i^{\ast }$ for the dual of $i$. Put $N_{ij}^k={\dim }_{\mathbb{C}}\mathrm{Hom}(k,i\otimes j)$, $d_i={\mathrm{tr}}_j(1_i)$, and $s_{ij}={\mathrm{tr}}_j(c_{j,i}{c}_{i,j})$. Define $D^2={\sum }_{i\in I}{d}_i^2$. The unnormalized Hopf-link matrix is $s=(s_{ij})$. The ribbon fusion category is modular when $s$ is invertible. When $D^2\mathrm{\ne }0$, either square root $D$ defines the normalized matrix $S=s\mathrm{/}D$; for a modular category, The modular Hopf-link matrix squares to D² times duality proves this nonvanishing. No positivity of the general $d_i$ or of $D$ is assumed.

The trace in $d_i$ and $s_{ij}$ uses the pivotal map $j=u\theta$ and the chosen left duals. Thus $s_{ij}$ is the trace of the double braiding on $i\otimes j$, not the trace of a bare endomorphism without a pivotal comparison. The labels and order in $s_{ij}={\mathrm{tr}}_j(c_{j,i}{c}_{i,j})$ are fixed throughout this pair.