16
votes
Accepted
Tannakian Formalism for the Quaternions and Dihedral Group
Let $V_D$ and $V_Q$ be the two dimensional simple representations of $D_4$ and $Q_8$ respectively. Let $1_D$ and $1_Q$ denote their trivial representations.
Suppose that there is a tensor equivalence ...
- 8,866
15
votes
Accepted
Is the category $\operatorname{sVect}$ an "algebraic closure" of $\operatorname{Vect}$?
$\newcommand\sVec{\mathrm{sVec}}\newcommand\Vec{\mathrm{Vec}}$Yes. Over an algebraically closed field of characteristic $0$, $\sVec$ is the algebraic closure of $\Vec$. By "algebraic closure" of $K$ I ...
- 54.6k
13
votes
Accepted
Why linearization leads to arithmetization?
I think:
The category of varieties over $\mathbb Q$ is already very arithmetic.
One reason that the linearization is considered arithmetic is that so much of the tractable arithmetic information is ...
- 148k
12
votes
Tannakian Formalism for the Quaternions and Dihedral Group
The categories ${\rm Rep}(Q_8)$ and ${\rm Rep}(D_8)$ are not equivalent as tensor categories. They have the same Grothendieck ring, but they have non equivalent associators. As far as I am aware, it ...
- 3,830
11
votes
Accepted
Derived version of equivalence between motives and representations of Motivic galois groups?
Let $k$ be a field and $\operatorname{DM}_{gm}(k)_{\mathbb Q}$ the ∞-category of rational geometric motives over $k$. A mixed Weil cohomology theory induces a symmetric monoidal exact functor
$$
R: \...
- 8,972
11
votes
Accepted
Are all representations of the geometric étale fundamental group subquotients of representations of the arithmetic étale fundamental group?
This is very far from true. In fact, a representation extending to the arithmetic fundamental group is such a severe restriction that these satisfy strong finiteness results; see [Litt21].
An easier (...
- 28.7k
10
votes
Accepted
Interpretation of the algebra of natural endomorphisms of the fiber functor of $\operatorname{Rep}(G)$
This is an elaboration on what is in the other answers. First, general categorical arguments can be used to prove the following. Let $K$ be a field and $A$ a $K$-algebra. The profinite completion $\...
- 118k
9
votes
Tannaka duality for semisimple groups
In order for ${\mathcal C}$ to come from an algebraic group rather than a pro-algebraic one, you want ${\mathcal C}$ to be finitely generated. And for semisimplicity, you want the group to have finite ...
- 615
9
votes
Interpretation of the algebra of natural endomorphisms of the fiber functor of $\operatorname{Rep}(G)$
For a reductive group the category of representations is semisimple so the algebra of endomorphisms of the fiber functor is just a product of matrix algebras, one for each irreducible representation.
...
- 148k
8
votes
Accepted
Tannaka duality for semisimple groups
Another criterion is that there should be only finitely many objects of bounded dimension. This condition might be easy to check in practice from abstract finiteness theorems. The proof is that, if ...
- 148k
8
votes
Accepted
A question about the Tannaka-Krein reconstruction of finite groups
I don't think you've made an error. There's a version of Tannakian reconstruction for merely monoidal categories with fiber functors. Namely, a finite dimensional Hopf algebra is the same thing as a ...
- 28.1k
8
votes
How much of the category of motives can be recovered from automorphisms of the Betti functor
A precise way to say that "sufficiently nice functors" from schemes to complexes of vector spaces determine realization functors of Voevodsky's motives is the notion of mixed Weil cohomology.
There ...
- 13.6k
6
votes
A question about the Tannaka-Krein reconstruction of finite groups
Although you already have one sufficient answer, I thought I would add here that Etingof and Gelaki, "Isocategorical groups" considered exactly this question. For finite groups, they gave a complete ...
- 3,230
6
votes
Interpretation of the algebra of natural endomorphisms of the fiber functor of $\operatorname{Rep}(G)$
In general I believe this should be the full dual algebra of $\mathcal{O}(G)$, or equivalently the profinite completion of the group algebra $k[G]$. If $G$ is simply connected, not necessarily ...
- 8,524
5
votes
Accepted
Question about references for proof of Proposition 1.3 in P. Deligne & J.S. Milne's article on "Tannakian Categories"
$\require{AMScd}$Let's prove that the left square commutes (I will only give a rough sketch). This means we want to prove that the following diagram commutes
$$
\begin{CD}
X\otimes Y @>l_{X\otimes ...
- 10.9k
5
votes
Tensor functor between rigid tensor categories preserves $\text{Hom}$-objects
Your $F$ preserves dual objects because it preserves the duality situations: quadruples $(X,Y,\alpha : X\otimes Y \rightarrow I, \beta: I \rightarrow Y \otimes X)$.
It remains to recall the formula ...
- 12.3k
5
votes
Why would the category of Motives be Tannakian?
My impression is that the language of Tannakian categories is a nice thing for itself; thus applying it to motives may (and will, see below) yield interesting applications. There are several ways of ...
- 16.9k
5
votes
Why linearization leads to arithmetization?
I'm not sure if this is already implicit in your question, but:
In the subsection "Artin motives" of this paper by Deligne and Milne, the authors explain how the reconstruction result Theorem 2.11 ...
- 944
5
votes
Accepted
What properties of a finite group fibre functor give its endomorphisms a hopf algebra structure?
The tensor product functor $\otimes : \text{Rep}(G) \times \text{Rep}(G) \to \text{Rep}(G)$ is "bilinear": it preserves colimits in both variables. It can therefore be regarded as being a "linear" ...
- 118k
5
votes
Accepted
Tannakian duality and the equivalence of $\alpha_p$- and $\mathbb{Z}/p\mathbb{Z}$-representation Categories
$\newcommand{\Z}{\mathbf{Z}}\DeclareMathOperator{\spec}{Spec}\DeclareMathOperator{\Rep}{Rep} \DeclareMathOperator{\QCoh}{QCoh}$The Cartier dual of $\alpha_p$ is itself, while the Cartier dual to $\Z/p$...
- 5,760
4
votes
Accepted
Tannakian formalism for topological Hopf algebras
I believe the best answer to your question at the moment is in this paper (which only treats the case of bialgebras)
http://adsabs.harvard.edu/abs/2014arXiv1411.3183L
Indeed, in topological setting ...
- 378
4
votes
Accepted
What is the name of the Hopf algebra whose comodules are the "positive" highest weight modules of $U_{q}(sl(2))$?
It usually goes by "quantized function algebra" of $SL_2$, or "regular function algebra" of $SL_q(2)$, of course with various minor variations. In Kassel's book it's denoted by $SL_q(2)$ (see Section ...
- 2,066
4
votes
Accepted
Functors between module categories that comes from restriction
Let's assume that we are using right modules and that the isomorphism of underlying vector space functors is a natural isomorphism because we will just work up to isomorphism of functors. Note that $...
- 38.6k
4
votes
Tannakian criterion for reducedness of Tannakian dual group
Yes, and one could argue as follows, at least if the group scheme is assumed to be of finite type (so it is an algebraic group), and the base field is perfect. Recall that if $k$ is a field of ...
- 5,760
3
votes
Tensor functor between rigid tensor categories preserves $\text{Hom}$-objects
$\DeclareMathOperator{\uphom}{\underline{Hom}}$
Looking at the notes, I can see that $\uphom(X,Y)$ is the internal hom of the category.
This has the universal property that that maps into it $ Z\to \...
- 221
3
votes
Accepted
Tannakian reconstruction for braided categories
Beware that the forgetful functor on the category of representations of a quasi-triangular Hopf algebra is typically not a braided functor. Rather, the general pattern is that you can reconstruct a ...
- 8,524
3
votes
Group action on fibre functor
Deligne proved that assuming that $C$ is Tannakian and $K$-linear, where $K$ is an algebraically closed field of characteristic zero, then there is a unique fiber functor from C to $Vec_K$. If you ...
- 5,039
3
votes
Functors on rigid tensor categories.
Here's an answer with all the relevant diagrams following Deligne's article "Catégories Tannakiennes" (look at 2.7 and 2.4). It gives the slightly stronger result that any natural ...
- 5,039
2
votes
Tannaka duality for semisimple groups
I've decided to turn my comments into an answer.
(a) The conditions characterizing the Tannakian categories attached to connected reductive groups can be found in Chapter 2 (2.20, 2.22, 2.23) of the ...
- 208
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