Tannakian categories equivalent as abelian categories - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T05:14:12Z http://mathoverflow.net/feeds/question/48033 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/48033/tannakian-categories-equivalent-as-abelian-categories Tannakian categories equivalent as abelian categories YBL 2010-12-02T11:11:03Z 2010-12-02T17:07:19Z <p>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 $H$? </p> <p>Question: Suppose $G$ is pro-unipotent. Are $G$ and $H$ isomorphic?</p> <p>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?</p>