Counter example in Tannaka reconstruction? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T07:33:35Z http://mathoverflow.net/feeds/question/98508 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/98508/counter-example-in-tannaka-reconstruction Counter example in Tannaka reconstruction? Bruce Westbury 2012-05-31T17:57:12Z 2012-05-31T17:57:12Z <p>This question is motivated by my attempts to answer the question <a href="http://mathoverflow.net/questions/96149" rel="nofollow">http://mathoverflow.net/questions/96149</a> from the point of view of Tannaka reconstruction. This has led me to the following problem and I am asking for an example (or alternatively an argument that what I am asking for does not exist).</p> <p>Let $K$ be a field and $G$ an algebraic group over $K$. I will use "tensor" as shorthand for "$K$-linear rigid symmetric monoidal". Then the category $Rep(G)$ of finite dimensional rational representations is a tensor category.</p> <p>Then let $C$ be a tensor category and $\omega\colon C\rightarrow Rep(G)$ a faithful tensor functor. There are two properties I am asking for. First, that $G$ is the group of tensor automorphisms of the functor from $C$ to vector spaces over $K$ (given by composing $\omega$ with the forgetful functor). Second that $\omega$ is <strong>not</strong> full.</p> <p>Note I am not assuming that $C$ is abelian. This is a standard condition. If this is assumed and $\omega$ is exact then Tannaka theory implies that $\omega$ is an equivalence. I also suspect (but can't prove) that if $G$ is reductive then $\omega$ is full.</p> <p>I would be particularly interested in an example with $K$ of characteristic zero (in which case $K$ is perfect and $G$ is reduced).</p>