Quantum dimension (${\dim }_q(V)={\mathrm{tr}}_q(1_V)$)

Closing an identity strand using compatible evaluation morphisms and coevaluation morphisms gives the categorical dimension in a pivotal or ribbon setting. For the standard two-dimensional quantum enveloping algebra of sl2 module, the weighted contraction has diagonal coefficients $q^{-1},q$, giving quantum dimension $q+q^{-1}=[2{]}_q$. The chosen duality and framing conventions matter.

The positive square root of the sum of the squared quantum dimensions of all simple anyons. The vacuum anyon has $d_1=1$ and antianyons have equal quantum dimensions.