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Suppose $A = Rep_k(G)$ and $B=Rep_k(H)$ are tannakian categories and $F: A\to B$ is an equivalence of abelian categories with $F(1_A) = 1_B$ (but not a $\otimes$-equivalence). What can I say about $G$ and $H$?

Question: Suppose $G$ is pro-unipotent. Are $G$ and $H$ isomorphic?

I have a vague feeling that there should some $H^1$ classifying deformations of the $\otimes$-structure on $A$ and that if $G$ is pro-unipotent this group should vanish so that $G \simeq H$. Is there any reference for such a thing?

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# Tannakian categories equivalent as abelian categories

Suppose $A = Rep_k(G)$ and $B=Rep_k(H)$ are tannakian categories and $F: A\to B$ is an equivalence of abelian categories with $F(1_A) = 1_B$ (but not a $\otimes$-equivalence). What can I say about $G$ and $H$?

I have a vague feeling that there should some $H^1$ classifying deformations of the $\otimes$-structure on $A$ and that if $G$ is pro-unipotent this group should vanish so that $G \simeq H$. Is there any reference for such a thing?