11
votes
1answer
359 views

Can one explain Tannaka-Krein duality for a finite-group to … a computer ? (How to make input for reconstruction to be finite datum?)

Consider a finite group. Tannaka-Krein duality allows to reconstruct the group from the category of its representations and additional structures on it (tensor structure + fiber functor). Somehow ...
12
votes
3answers
1k views

History and motivation for Tannaka, Krein, Grothendieck, Deligne et al. works on Tannaka-Krein theory?

I am trying to wrap my mind around Tannaka-Krein duality and it seems quite mysterious for me, as well, as its history. So let me ask: Question: What was the motivation and historical context for ...
1
vote
2answers
289 views

In which categories is every coalgebra a sum of its finite-dimensional subcoalgebras?

Let $\mathcal V$ be a reasonably nice category — I'm interested in the case when $\mathcal V$ is $\mathbb K$-linear for some field $\mathbb K$, abelian, and has all products and coproducts ...
7
votes
3answers
648 views

Tannakian description of a semi-direct product

Let $H$ and $K$ be affine group schemes over a field $k$ of caracteristic zero. Let $\varphi:H\to Aut(K)$ be a group action. Then we can form the semi-direct product $G = K\ltimes H$. Problem: ...
9
votes
3answers
771 views

When does Tannakian theory work over affine schemes besides fields?

By 'work' I would like the correspondence between fiber functors (to finitely generated projective modules) and algebraic groups to be the same as in the field case. Specifically, if $A$ is an affine ...
3
votes
3answers
418 views

What is a formula for the “group-like Drinfeld element”?

Any quantized universal enveloping algebra (in fact, any toplogically quasi-triangular Hopf algebra) has an (in its completion) an element u called the Drinfeld element which gives an isomorphism from ...
4
votes
2answers
402 views

Are there interesting monoidal structures on representations of quantum affine algebras?

Is there a good monoidal structure on a category of integrable representations of a quantum affine algebra? In the ordinary affine Kac-Moody case, there is the usual tensor product (symmetric, adds ...