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Jeremy Rickard
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How wierdweird can Modular Tensor Categories be over non-algebraically closed fields?

I am trying to understand better the behaviour and character of modular tensor categories over non-algebraically closed fields. How weird can they be?

The reason I am interested in this is that my collaborators and I are investigating extended 3D tqfts. These are known to be related to Modular Tensor Categories via the Reshetikhin-Turaev construction. However most of the literature seems to be focused on the case where the MTC is defined over an algebraically closed field, usually the complex numbers.

I don't see a reason to restrict to this case and I can imagine that there could be some very interesting examples, and hence interesting invariants, in other cases to.

For example over field $k$, a finite semisimple linear category will have simple objects whose endomorphisms rings are division algebras over k$k$. If k$k$ is algebraically closed, then all we get are copies of k$k$. If k$k$ is not algebraically closed then it is more interesting as we can have objects with different division algebras as endomorphisms.

Can this happen in a modular tensor category? i.e. is there an example of a Modular Tensor Category such that the simple objects have different divisionaldivision algebras for their endomorphism rings? What if we drop the requirement End(1) = k?

How sticky can it get?

How wierd can Modular Tensor Categories be over non-algebraically closed fields?

I am trying to understand better the behaviour and character of modular tensor categories over non-algebraically closed fields. How weird can they be?

The reason I am interested in this is that my collaborators and I are investigating extended 3D tqfts. These are known to be related to Modular Tensor Categories via the Reshetikhin-Turaev construction. However most of the literature seems to be focused on the case where the MTC is defined over an algebraically closed field, usually the complex numbers.

I don't see a reason to restrict to this case and I can imagine that there could be some very interesting examples, and hence interesting invariants, in other cases to.

For example over field $k$, a finite semisimple linear category will have simple objects whose endomorphisms rings are division algebras over k. If k is algebraically closed, then all we get are copies of k. If k is not algebraically closed then it is more interesting as we can have objects with different division algebras as endomorphisms.

Can this happen in a modular tensor category? i.e. is there an example of a Modular Tensor Category such that the simple objects have different divisional algebras for their endomorphism rings? What if we drop the requirement End(1) = k?

How sticky can it get?

How weird can Modular Tensor Categories be over non-algebraically closed fields?

I am trying to understand better the behaviour and character of modular tensor categories over non-algebraically closed fields. How weird can they be?

The reason I am interested in this is that my collaborators and I are investigating extended 3D tqfts. These are known to be related to Modular Tensor Categories via the Reshetikhin-Turaev construction. However most of the literature seems to be focused on the case where the MTC is defined over an algebraically closed field, usually the complex numbers.

I don't see a reason to restrict to this case and I can imagine that there could be some very interesting examples, and hence interesting invariants, in other cases to.

For example over field $k$, a finite semisimple linear category will have simple objects whose endomorphisms rings are division algebras over $k$. If $k$ is algebraically closed, then all we get are copies of $k$. If $k$ is not algebraically closed then it is more interesting as we can have objects with different division algebras as endomorphisms.

Can this happen in a modular tensor category? i.e. is there an example of a Modular Tensor Category such that the simple objects have different division algebras for their endomorphism rings? What if we drop the requirement End(1) = k?

How sticky can it get?

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Chris Schommer-Pries
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How wierd can Modular Tensor Categories be over non-algebraically closed fields?

I am trying to understand better the behaviour and character of modular tensor categories over non-algebraically closed fields. How weird can they be?

The reason I am interested in this is that my collaborators and I are investigating extended 3D tqfts. These are known to be related to Modular Tensor Categories via the Reshetikhin-Turaev construction. However most of the literature seems to be focused on the case where the MTC is defined over an algebraically closed field, usually the complex numbers.

I don't see a reason to restrict to this case and I can imagine that there could be some very interesting examples, and hence interesting invariants, in other cases to.

For example over field $k$, a finite semisimple linear category will have simple objects whose endomorphisms rings are division algebras over k. If k is algebraically closed, then all we get are copies of k. If k is not algebraically closed then it is more interesting as we can have objects with different division algebras as endomorphisms.

Can this happen in a modular tensor category? i.e. is there an example of a Modular Tensor Category such that the simple objects have different divisional algebras for their endomorphism rings? What if we drop the requirement End(1) = k?

How sticky can it get?