Questions tagged [tannakian-category]
The tannakian-category tag has no usage guidance.
99
questions
6
votes
0
answers
166
views
Is there a finite test for isomorphisms of rigid monoidal abelian categories, part II
Let $G$ be a semisimple algebraic group. (I'm already interested in the case $G=SL_2$.)
Let $\mathcal C$ be a semisimple rigid monoidal abelian category endowed with an additive tensor functor $Rep_G ...
12
votes
1
answer
252
views
Is there a finite test for isomorphisms of rigid monoidal abelian categories?
Let $G$ be a semisimple algebraic group. (I'm already interested in the case $G=SL_2$.)
Let $\mathcal C$ be a semisimple rigid monoidal abelian category endowed with pair of exact tensor functors $...
4
votes
1
answer
492
views
Faithful exact functors to tensor categories
Let $P$ be a "nice" $k$-linear abelian tensor category (e.g. A tannakian category or a fusion category over a field $k$) and $F: M\to P $ an additive $k$-linear exact and faithful functor. I want to ...
1
vote
1
answer
156
views
Galois group - unknown word
I found in some articles a definition of the Galois group of a differential module. In the definition appeared the expression
$(Repr(Gal(M,\nabla),Forget)$. According to the the articles, this pair ...
11
votes
0
answers
660
views
What is the role of fiber functor in Deligne's theorem on Tannakian categories?
The theorem states that, for a field $k$ of characteristic 0, any $k$-linear tensor category with $End(1)=k$ satisfying a condition that each object is annihilated by a Schur functor, is equivalent to ...
11
votes
3
answers
756
views
Exact sequences of groups and Tannakian formalism
By work of Deligne and others (I am following Deligne-Milne's notes which I just began to read: http://www.jmilne.org/math/xnotes/tc.pdf) we know that a given affine group scheme G can be recovered ...
12
votes
2
answers
1k
views
k-linear abelian categories which are not categories of modules
According to Joyal, Street ("An Introduction to Tannaka Duality and Quantum Groups"), any $k$-linear abelian category $\mathcal{C}$ admitting a faithful, exact functor $U: \mathcal{C} \rightarrow \...
57
votes
0
answers
3k
views
Grothendieck's Period Conjecture and the missing p-adic Hodge Theories
Singular cohomology and algebraic de Rham cohomology are both functors from the category of smooth projective algebraic varieties over $\mathbb Q$ to $\mathbb Q$-vectors spaces. They come with the ...
6
votes
1
answer
324
views
Establishing Duality in Tannakian Categories
I sometimes need to prove a category is Tannakian. Part of the definition of a Tannakian category is that it is rigid.
However, I find the definition of rigid categories somewhat difficult. I don't ...
6
votes
1
answer
711
views
Tannakian fundamental group of two explicit tensor categories
Let $K/k$ is a field extension and $G$ an affine group scheme over $K$. What are the Tannakian fundamental groups of these two $k$-tensor categories (with trivial fiber functors over $k$):
1. The ...
7
votes
0
answers
1k
views
Torsors and twists of algebraic groups
Let $G/S$ be an affine group scheme. Then the automorphism group of every $G$-torsor over $S$ is a twist of $G$, but it this functor isn't essentially surjective in general (It may be not full nor ...
4
votes
1
answer
420
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 ...
10
votes
1
answer
526
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 Etingof-...
4
votes
0
answers
318
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
votes
0
answers
198
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 ...
9
votes
3
answers
730
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 $k$)...
0
votes
3
answers
390
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 ...
30
votes
1
answer
2k
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 ...
3
votes
0
answers
177
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 ...
5
votes
2
answers
673
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 ...
13
votes
0
answers
871
views
Stack of Tannakian categories? Galois descent?
I'm having trouble finding a reference for something that I'm guessing the experts worked out long ago. Let's take a local or global field $F$ for this post, and fix a separable algebraic closure $\...
7
votes
1
answer
469
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 ...
3
votes
0
answers
304
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
1
answer
639
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
...
5
votes
1
answer
468
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 ...
3
votes
1
answer
1k
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
0
answers
351
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
votes
2
answers
502
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 \mathrm{GL}...
12
votes
1
answer
872
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 $G$-...
26
votes
2
answers
2k
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 up,...
2
votes
2
answers
1k
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 ...
10
votes
1
answer
2k
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) ...
23
votes
1
answer
3k
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 ...
8
votes
2
answers
1k
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 "...
23
votes
2
answers
2k
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 ...
8
votes
1
answer
1k
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. ...
16
votes
2
answers
3k
views
The algebraic version of Riemann-Hilbert correspondence
It is well known that if I have a differentiable manifold (holomorphic manifold) $M$, then I have a functor from the category of vector bundles on $M$ with flat connections to the category of local ...
7
votes
3
answers
1k
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 ...
8
votes
1
answer
982
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
1
answer
764
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
1
answer
491
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 ...
6
votes
0
answers
405
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 $G$ ...
19
votes
2
answers
4k
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 (...
18
votes
1
answer
1k
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
1
answer
810
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 ...
44
votes
2
answers
3k
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 ...
15
votes
1
answer
708
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 \...
9
votes
3
answers
1k
views
Does any tensor category correspond to a bialgebra?
I wonder how strong the power of Tannaka philosophy is, and if we accept that a tensor category is a generalized bialgebra, what difficulties we will come up against ?
Edit: Whether most tensor ...
40
votes
4
answers
7k
views
Tannakian Formalism
The Tannakian formalism says you can recover a complex algebraic group from its category of finite dimensional representations, the tensor structure, and the forgetful functor to Vect. Intuitively, ...