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2
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93 views

Applying Lemma 2.12 of Deligne's/ Milne's “Tannakian Categories” on an irreducible representation

since this is my first question here, I'm not very certain whether I state it properly. I'm thankful for any helpful remarks. Currently I'm trying to understand the above-mentioned article which can ...
5
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0answers
143 views

Tannakian formalism for topological Hopf algebras

Tannaka-Krein duality allows, under the appropriate assumptions, to reconstruct a Hopf algebra from its category of modules. This method was found to be powerful for instance in the work of ...
4
votes
0answers
169 views

Tannaka categories and reductive groups

The group associated to a Tannaka category $T$ over a field is pro-reductive if and only if $T$ is semi-simple. Pro-reductive groups make sense over any scheme. Is there an extension of the theory ...
5
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0answers
111 views

Algebraic fundamental group without regularity at infinity

Suppose $X$ is a smooth (connected) variety over $\mathbb{C}$. Let $\mathscr{C}$ be the category of finite rank vector bundles on $X$ equipped with an integrable connection, and let $\mathscr{C}'$ be ...
5
votes
2answers
263 views

Generalized Tannakian Duality?

By the classical theory of Tannakian duality we know that every $k$-linear rigid abelian tensor category ($k$ a field) which has a fibre functor to $\mathrm{Vec}_k$ (finite dim. vector spaces over ...
0
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3answers
277 views

Rank vanishing in tensor categories

Let $\mathcal{C}$ be a cocomplete $\mathbb{Q}$-linear symmetric monoidal category (if convenient, assume that it is locally finitely presentable and/or abelian). Recall that the rank (or dimension) of ...
11
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1answer
398 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 ...
2
votes
0answers
116 views

quantum deformations of tensor category

I was told that, if I understand correctly, that the enveloping algebra of semisimple Lie algebra admits one family of quantum deformation as Hopf algebra, which was proved by Drinfeld. Anyone can ...
4
votes
1answer
245 views

Is a group scheme determined by its category of representations?

More precisely, let $G$ be an affine group scheme over a field $k$, $Rep_k(G)$ be the category of finite dimensional representations of G, and $\omega_0$ be the forgetful functor from $Rep_k(G)$ to ...
6
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1answer
245 views

About an embedding of abelian categories into categories of modules

Let $k$ be a field. Let $C$ be an abelian $k$-linear category with a symmetric tensor product $\otimes$ and internal homomorphisms, such that $\mathrm{End}(1)=k$. Let $M$ be another $k$-linear abelian ...
1
vote
0answers
127 views

When does Ext^2 vanish in a category of group representations.

Let $G$ be a linear algebraic group over field $k$ of characteristic zero. It is well known that the category of finite dimensional $k$--linear representations of $G$ is abelian, and that it is ...
2
votes
1answer
364 views

Tannaka Duality

I only know two theorems for neutral Tannaka categories. (1) One states that the set of $k$-group scheme homomorphisms $m:G_1\to G_2$ is in one to one correspondence with the set of $k$-linear ...
4
votes
1answer
199 views

Functors on rigid tensor categories.

This is a question about the proof of proposition 1.13 in Deligne and Milne, Tannakian Categories. Let $C,C'$ be two rigid tensor categories and $F,G : C \rightarrow C'$ be two tensor functors. Let $u ...
0
votes
1answer
242 views

automorphism of fibre functors

If $G$ is an affine group scheme over a field $k$, then we have the forgetful functor $\omega$ from the category ${\rm Rep}_k(G)$ of finite representations of $G$ to the category of finite dimensional ...
4
votes
0answers
258 views

Counter example in Tannaka reconstruction?

This question is motivated by my attempts to answer the question Invariants for the exceptional complex simple Lie algebra $F_4$ from the point of view of Tannaka reconstruction. This has led me to ...
9
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2answers
329 views

Are algebraic groups defined by their invariants in tensor spaces?

Let $K$ be a field of characteristic zero, and let $G \subseteq \mathrm{GL}_V$ be an algebraic group over $K$, acting faithfully on a finite dimensional vector space $V$. Let $H \subseteq ...
11
votes
1answer
654 views

Fiber functors to derived categories

Suppose that $G$ is an algebraic group over a field $k$. Then for any $k$-algebra $R$, a fiber functor from $\text{Rep}_k(G)$ to the category of projective modules over $R$ is the same as a ...
21
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1answer
898 views

What does the Tannakian formalism reconstruct when fed the category of chain complexes?

I've recently realized that there is a gap in my understanding of the Tannakian formalism for reconstructing an (algebraic) group from its category of (finite-dimensional) representations. To warm ...
2
votes
1answer
539 views

isomorphism of fibre functors

If $\mathfrak{C}$ is a $k$-linear rigid abelian tensor category with End(1)=$k$(strictly speaking is isomorphic to $k$ as a $k$-algebra), and $k=\bar{k}$, and if $\omega_1$ and $\omega_2$ are two ...
7
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1answer
652 views

How does the conjectural Langlands group fit into the Tannakian point of view?

I've read that one way to formulate the Langlands program is the following: Let $\mathcal{L}_ {\mathbb{Q}}$ be the conjectural Langlands group. Then the category of semi-simple (continuous) ...
14
votes
1answer
1k views

What is the precise relationship between Langlands and Tannakian formalism?

As anyone who's been reading the forums closely can see, I've been averaging a question a day about Tannakian formalism for the past few days. It's quite an interesting concept! In any case, I wish ...
7
votes
2answers
465 views

Is there a ``path'' between any two fiber functors over the same field in Tannakian formalism?

I will take the approach of this question: Tannaka formalism and the étale fundamental group and think of the etale fundamental group as Tannakian formalism for $\mathbb{F}_1$. Then our ...
10
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1answer
799 views

Why would the category of Motives be Tannakian?

After reading the answer to my previous question: What are the different theories that the motivic fundamental group attempts to unify? I decided to read up on Tannakian formalism. Given the ...
7
votes
1answer
585 views

What extra conditions are necessary for the following version of Koszul duality?

Conventions: So that I don't have to worry about, fix a field $k$ of characteristic zero, and always work over it. Categories of modules, etc., are always $\infty$-categories of dg modules. ...
11
votes
2answers
1k views

The algebraic Version of Riemann Hilbert Correspondence

It is well known that if I have a differentialable manifold (holomorphic maniford) $M$, then I have a functor from the categroy of vector bundles on $M$ with flat connections to the categroy of local ...
5
votes
3answers
806 views

For quasi-coherent D-Modules

It is well-know that the category of coherent D-modules over a smooth algebraic $k$-scheme is a Tannakian category. So it is equivalent to the category of finite representations of some affine group ...
7
votes
1answer
806 views

commutativity constraint in Grothendieck's motives

This is a basic question about Grothendieck's conjectural category $M_k$ of pure motives (over a field $k$). This construction first produces a category (the "false category of motives") which need ...
4
votes
1answer
578 views

Mumford-Tate groups and Hodge structures

It is well known that the Mumford-Tate group of a polarizable pure $Q$-Hodge structure is reductive. (this is proved for instance in Deligne et al LNM 900, Voisin's books on Algebraic Geometry,..) ...
2
votes
1answer
370 views

Hurewicz theorem related to Galois group (or Tannakian categories)?

Is there a proof of the Hurewicz theorem $\pi_1(X)^{ab} = H_1(X, \mathbf Z)$ ($X$ a connected topological space) expressing $\pi_1(X)$ as the "Galois" group of $X$, i.e., group of deck transformations ...
4
votes
0answers
300 views

Tannakian categories equivalent as abelian categories

Suppose $A = Rep_k(G)$ and $B=Rep_k(H)$ are tannakian categories and $F: A\to B$ is an equivalence of abelian categories with $F(1_A) = 1_B$ (but not a $\otimes$-equivalence). What can I say about $G$ ...
11
votes
2answers
2k views

What's so special about the forgetful functor from G-rep to Vect?

The following is some version of Tannaka-Krein theory, and is reasonably well-known: Let $G$ be a group (in Set is all I care about for now), and $G\text{-Rep}$ the category of all $G$-modules ...
8
votes
1answer
598 views

Does the Tannaka-Krein theorem come from an equivalence of 2-categories?

Possibly the correct answer to this question is simply a pointer towards some recent literature on Tannaka-Krein-type theorems. The best article I know on the subject is the excellent André Joyal ...
5
votes
1answer
633 views

Fiber functor of category of D-module on affine Grassmannian.

Geometric Satake correspondence allows us to construct Langlands dual group in a canonical way. In Mirkovic-Vilonen's paper, they prove that category of spherical perverse sheaves is an commutative ...
34
votes
2answers
1k views

What algebraic group does Tannaka-Krein reconstruct when fed the category of modules of a non-algebraic Lie algebra?

Let $\mathfrak g$ be a finite-dimensional Lie algebra over $\mathbb C$, and let $\mathfrak g \text{-rep}$ be its category of finite-dimensional modules. Then $\mathfrak g\text{-rep}$ comes equipped ...
14
votes
1answer
481 views

Semistable filtered vector spaces, a Tannakian category.

Let $k$ be a field (char = 0, perhaps). Let $(V,F)$ be a pair, where $V$ is a finite-dimensional $k$-vector space, and $F$ is a filtration of $V$, indexed by rational numbers, satisfying: $F^i V ...