Reconstruction of tensor categories of type $G_2$
Lilit Martirosyan, Hans Wenzl
TL;DR
The paper proves a complete reconstruction and classification for non-symmetric ribbon tensor categories of type $G_2$ and its level-$k$ analogs $G_{2,k}$, showing they are equivalent to $ ext{Rep }U_q( rak g(G_2))$ with $q^2$ not a root of unity (and to the corresponding quotient categories $ar{U}_q$ at appropriate roots of unity). The authors develop a braid-representation–based program: analyze $B_n$ actions on the fundamental 7-dimensional generating object $V$, classify simple $K_4$-modules, use Temperley–Lieb algebras to control eigenvalues of $c_{V,V}$, and prove surjectivity of braid representations generating ${ m End}(V^{ ensor n})$ for all $n$; this, together with reconstruction techniques, yields the desired equivalences. They provide two perspectives: (i) a concrete braid-eigenvalue/path approach and (ii) an alternative trivalent/spider–theoretic proof following Kuperberg’s framework, linking to the graphical calculus of the $G_2$ spiders. The results connect to known fusion categories (e.g., Fibonacci, $SO(3)_7$) and pave the way for extending the classification to other exceptional Lie types via braid and spider methods, including a broader view of roots of unity and their impact on semisimplicity and dimensions.
Abstract
We prove that any non-symmetric ribbon tensor category $\mathcal{C}$ with the fusion rules of the compact group of type $G_2$ needs to be equivalent to the representation category of the corresponding Drinfeld-Jimbo quantum group for $q$ not a root of unity. We also prove an analogous result for the corresponding finite fusion tensor categories.