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G-Reconstruction and the Hopf Equivariantization Theorem
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- Author / Creator
- Riesen, Andrew
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$G$-structures on fusion categories have been shown to be an important tool to understand orbifolds of vertex operator algebras \cite{Kirillov}\cite{Gcrossedmuger}\cite{Orbifold_Paper}. We continue to develop this idea by generalizing Eilenberg-Maclane's notion of an Abelian $3$-cocycle to describe $G$-structures on fusion categories as $G$-(crossed, ribbon) Abelian $3$-cocycles on an algebra $H$. In particular, we show that a $G$-(crossed, ribbon) Abelian $3$-cocycle on $H$ will induce a $G$-(crossed braided, ribbon) tensor structure on its category of modules $\mathrm{Mod}(H)$. We then prove that every $G$-(crossed braided, ribbon) fusion category $\mathcal{C}$ will be equivalent to the category of modules of some finite dimensional algebra $H$ with $G$-structure induced from a $G$-(crossed, ribbon) Abelian $3$-cocycle. We call this $G$-reconstruction.
Lastly, we prove that a $G$-ribbon Abelian $3$-cocycle $\Gamma$ on $H$ allows us to describe the equivariantization $(\Mod(H))^G$ as the category of modules of a ribbon (weak) quasi Hopf algebra $H\#_{\Gamma}\mathbb{C}[G]$. We call this the Hopf equivariantization theorem.
By $G$-reconstruction this shows that if $\mathcal{V}$ is a strongly rational vertex operator algebra where $G$ acts faithfully on $\mathcal{V}$ such that $\mathcal{V}^G$ is also strongly rational, then there is an equivalence of modular fusion categories:
\begin{equation}
\mathrm{Mod}\ \mathcal{V}^G \cong \mathrm{Mod}(H\#_{\Gamma}\mathbb{C}[G])
\end{equation}
for some finite dimensional algebra $H$ with a $G$-ribbon Abelian $3$-cocycle $\Gamma$. This provides a proof of the Dijkgraaf-Witten conjecture, and generalizes it as far as possible in the semi-simple setting. -
- Graduation date
- Fall 2023
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- Type of Item
- Thesis
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- Degree
- Master of Science
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- License
- This thesis is made available by the University of Alberta Libraries with permission of the copyright owner solely for non-commercial purposes. This thesis, or any portion thereof, may not otherwise be copied or reproduced without the written consent of the copyright owner, except to the extent permitted by Canadian copyright law.