Skein-Theoretic Methods for Unitary Fusion Categories
- URL: http://arxiv.org/abs/2008.07129v3
- Date: Wed, 5 May 2021 04:03:00 GMT
- Title: Skein-Theoretic Methods for Unitary Fusion Categories
- Authors: Anup Poudel and Sachin J. Valera
- Abstract summary: Unitary fusion categories (UFCs) have gained increased attention due to emerging connections with quantum physics.
We consider a fusion rule of the form $qotimes cong mathbf1oplusbigoplusk_i=1x_i$ in a UFC $mathcalC$, and extract information using the graphical calculus.
- Score: 0.0
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: Unitary fusion categories (UFCs) have gained increased attention due to
emerging connections with quantum physics. We consider a fusion rule of the
form $q\otimes q \cong \mathbf{1}\oplus\bigoplus^k_{i=1}x_{i}$ in a UFC
$\mathcal{C}$, and extract information using the graphical calculus. For
instance, we classify all associated skein relations when $k=1,2$ and
$\mathcal{C}$ is ribbon. In particular, we also consider the instances where
$q$ is antisymmetrically self-dual. Our main results follow from considering
the action of a rotation operator on a "canonical basis". Assuming self-duality
of the summands $x_{i}$, some general observations are made e.g. the
real-symmetricity of the $F$-matrix $F^{qqq}_q$. We then find explicit formulae
for $F^{qqq}_q$ when $k=2$ and $\mathcal{C}$ is ribbon, and see that the
spectrum of the rotation operator distinguishes between the Kauffman and
Dubrovnik polynomials.
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