Timeline for 2-TQFT are to Frobenius Algebras as ??? are to Hopf Algebras
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| Mar 10, 2017 at 9:42 | history | edited | CommunityBot |
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| Jun 22, 2012 at 2:25 | comment | added | David Jordan | Hrm, I think I retract the assertion that finite dimensional semi-simple Hopf algebras are pivotal. This is an open question for fusion categories; I thought it was known for ss Hopf algebras, but it appears not. In any case, one still has V\cong V^**, but not necessarily an isomorphism of monoidal functors id --> -^**, which still mimics the Frobenius algebra setting (lets you have Hom(1,ab)=Hom(1,ba), for instance). | |
| Jun 22, 2012 at 2:17 | comment | added | David Jordan | spaces, and so are not really intrinsic (from this point of view) of course they are crucial in constructing new examples, and relating to theory of algebraic groups. | |
| Jun 22, 2012 at 2:15 | comment | added | David Jordan | ... of Frobenius algebras (just as Frobenius algebras have a trace, pivotal fusion categories C have a "trace" Hom(1,-): C-->Vect, where 1 denotes the tensor unit in C (for a Hopf algebra the trivial representation). So rather than try to understand the axioms of Hopf algebras in TFT language, it is more fruitful to recognize that the categories of representations of Hopf algebras are essentially a categorification of Frobenius algebras, and thus lead to 3D TQFT's. I would emphasize, too, that Hopf algebras arise from this point of view as the automorphisms of a given functor to vector.. | |
| Jun 22, 2012 at 2:12 | comment | added | David Jordan | I just came upon this post, and was surprised not to find a discussion of the recent work of Douglas, Schommer-Pries, and Snyder. They explain that the fully dualizable objects in the 3-category whose objects are fusion categories, 1-morphisms are bimodule categories, 2-morphisms are bimodule functors, and 3-morphisms are natural transformations, are precisely fusion categories. Examples of fusion categories come from the representations categories of finite dimensional semi-simple Hopf algebras, and in this case the fusion categories you get are pivotal, which is a good categorification... | |
| Jun 22, 2012 at 1:49 | comment | added | Carlos Segovia | Hello Do you check this article arxiv.org/pdf/hep-th/9412025.pdf It is a different approach, if you sen me your mail I can send you an exposition I prepared about this article. Best Carlos my mail is [email protected] | |
| May 25, 2011 at 16:39 | answer | added | fosco | timeline score: 4 | |
| May 25, 2011 at 14:13 | comment | added | B. Bischof | Super interesting question | |
| May 25, 2011 at 11:51 | vote | accept | fosco | ||
| May 25, 2011 at 1:53 | answer | added | Theo Johnson-Freyd | timeline score: 8 | |
| May 25, 2011 at 0:28 | answer | added | Charlie Frohman | timeline score: 4 | |
| May 24, 2011 at 22:05 | history | edited | fosco | CC BY-SA 3.0 |
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| May 24, 2011 at 21:25 | comment | added | fosco | What do you mean by that? | |
| May 24, 2011 at 20:50 | comment | added | Qiaochu Yuan | I have seen what the relationship between the product and coproduct looks like in a bialgebra written out diagrammatically, and it just isn't very topological... | |
| May 24, 2011 at 20:14 | history | edited | fosco | CC BY-SA 3.0 |
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| May 24, 2011 at 20:06 | history | asked | fosco | CC BY-SA 3.0 |