what’s n in E_n?
In “factorization algebras = E_n algebras” does the curve also need to be A^1
The statement in topology is that translation-invariant factorization cosheaves on R^n are the same as E_n-algebras.
Does anybody know if Lurie's theorem is still true if you replace Rep_q(T) by some other braided monoidal category?
Lurie’s theorem is the combination of two things: (1) E_2 vs factorization on R^2 (2) Drinfeld center vs E_2. The second part can be generalized to any ambient presentable E_2-category.
So the answer is yes, right?
Sasha, in a sense, morally yes, but the point of using that category is that you have a good notion of positivity. Essentially, you want to use positivity to get simple Koszul duality statements (this is where Hopf algebras appear: you Koszul dualize the coalgebra structure to get an E_2-algebra). So it wouldn’t work as beautifully.
(t-exact at negative level)
I have a particular example in mind - in that case Rep_q(T) is replaced by Rep_q(GL(n)\times GL(m)) and some particular Hopf algebra there
That distinction can be absorbed into the definitions and is a little more subtle.
The convolution in the coherent RHS should, I guess, correspond to some kind of fusion…
let’s discuss this!
in fact, if you think of LocSys, this convolution has no meaning
Convolution doesn’t preserve the support condition….
Sam, yeah, maybe restrict to support at 1…
Dennis, why—doesn’t the factorization str give us the fusion over C?
Isn't it true that under the iso G^/G = LS^, convolution in QCoh goes over to fusion?
that’s what I’d think…
I don’t think so, but maybe am wrong
If we use Betti local systems, then we have LS = Map(S^1,pt/G) = G/G and I thought the E_2 structure on the LHS corresponds to convolution on the RHS