bio | website | sbseminar.wordpress.com |
---|---|---|
location | Bloomington, Indiana | |
age | 34 | |
visits | member for | 4 years, 10 months |
seen | 5 hours ago | |
stats | profile views | 6,003 |
Assistant Professor at Indiana University, working on tensor categories and their relationships to operator algebras and topology.
Jul 16 |
comment |
How weird can Modular Tensor Categories be over non-algebraically closed fields?
There's a bunch of great stuff related to Andre's comment in this short paper of Greg's. |
Jul 16 |
comment |
How weird can Modular Tensor Categories be over non-algebraically closed fields?
Modularity should be thought of as being as far from symmetry as possible. In the symmetric case the S-matrix has rank 1, and in the modular case it has full rank. On the other hand, Rep(G) is contained in its center which is modular, so @DavidSpeyer's example still works. |
Jul 6 |
comment |
When is/isn't the monoidal unit compact projective?
Very minor point, for A-mod-A projectivity is separability of A which in enough generality is stronger than semisimplicity (though they agree for algebras over a perfect field). |
Jul 6 |
answered | Rank vanishing in tensor categories |
Jul 2 |
awarded | Curious |
Jul 2 |
awarded | Enlightened |
Jul 2 |
awarded | Nice Answer |
Jul 1 |
revised |
How do we handle the symmetry condition in nCob and TQFTs?
typo |
Jul 1 |
comment |
How do we handle the symmetry condition in nCob and TQFTs?
Any symmetric tensor category can be strictified, so there's certainly some version of the bordism category where $M \sqcup N = N \sqcup M$. Scrictification is not a very natural thing to do though, and Oscar's right that with standard definitions $M \sqcup N \neq N \sqcup M$. |
Jul 1 |
answered | How do we handle the symmetry condition in nCob and TQFTs? |
Jun 27 |
answered | Strictifying strong monoidal functors |
Jun 27 |
awarded | Good Answer |
Jun 24 |
comment |
Is every separable algebra in a modular tensor category Morita equivalent to a commutative one?
It's worth noting that A being commutative corresponds to mod-A being an E_2-module category over C. Part of being an E_2-module category is being a tensor category in a compatible way, but another part is additional compatibility in the braiding. My first paragraph is about situations where the first part breaks, but for this SVec example the first part is fine but it's still not a commutative algebra (by inspection) and so it's the second part that fails. |
Jun 24 |
comment |
Is every separable algebra in a modular tensor category Morita equivalent to a commutative one?
The module category is Vec, so there are no non-free modules for that algebra. |
Jun 24 |
revised |
Is every separable algebra in a modular tensor category Morita equivalent to a commutative one?
added 48 characters in body |
Jun 24 |
comment |
Is every separable algebra in a modular tensor category Morita equivalent to a commutative one?
Good point. Yes, I meant the former. |
Jun 24 |
comment |
Is every separable algebra in a modular tensor category Morita equivalent to a commutative one?
There's a forgetful functor from SVec to Vec, so internal endomorphisms of the 1-dimensional vector space gives a 2-dimensional algebra in SVec. Since Vec is semisimple this is separable. Indeed it's $1 \oplus F$ and there's only one algebra structure on it. |
Jun 24 |
comment |
Is every separable algebra in a modular tensor category Morita equivalent to a commutative one?
I'm not sure about your second question, since symmetric is pretty restrictive. You can't get a counterexample from su(2). It seems implausible to me that the answer could be yes, but it also might be hard to find a counterexample. |
Jun 24 |
revised |
Is every separable algebra in a modular tensor category Morita equivalent to a commutative one?
deleted 103 characters in body |
Jun 24 |
comment |
Is every separable algebra in a modular tensor category Morita equivalent to a commutative one?
Hopefully my edit clarified your second question. The issue is whether the module category has a tensor product that's compatible with the module category structure. |