# Tagged Questions

**2**

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28 views

### From the representation category of a Lie group and the representation on a homogeneous space, can we reconstruct the stabiliser subgroup reps?

Given a Lie group $G$ and a transitive action $- \triangleright - : G \times X \to X$ on a homogeneous space, we can recover the stabiliser subgroup $H_x$ of a point $x \in X$. It is the subgroup of ...

**10**

votes

**1**answer

321 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

**3**answers

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 ...

**13**

votes

**2**answers

365 views

### Does base extension reflect the property of being isomorphic?

Let L/K be a (separable?) field extension, let A be a finite dimensional algebra over K, and let M and N be two A-modules. Let $A' = L \otimes_K A$ be the algebra given by extension of scalars, and ...

**4**

votes

**1**answer

229 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 ...

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vote

**0**answers

127 views

### Is this a pure monomorphism?

Let $M$ be a representation of a quiver $Q=(V, E)$ by $R$-modules. By $M^{+}$ we mean a representation of $Q^{op}$ with $M^{+}(v)=\mathrm{Hom}(M(v), \frac{Q}{Z})$. One can easily see that there is ...

**8**

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**3**answers

436 views

### What is the categorical significance of the trivial $\mathfrak{g}$-module in the category of $\mathfrak{g}$-mod?

This question may be trivial for experts.
Let $\mathfrak{g}$ be a Lie algebra over a field $k$ and consider the category $\mathfrak{g}$-mod of $\mathfrak{g}$ modules. We can add suitable conditions, ...

**4**

votes

**1**answer

194 views

### U(1) vs. BZ and representations of 2-groups

$U(1)$ seems to lead a dual life. On one hand it is the group we know and love, and on the other, it is the classifying space of the integers. Thinking about $n$-groups says that we should also think ...

**14**

votes

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461 views

### An algebraic approach to the thermodynamic limit $N\rightarrow\infty$?

In physics one studies quite often the thermodynamic limit or what we call the $N\rightarrow \infty$ behavior of a system of $N\rightarrow\infty$ particles. This is of particular relevance in the ...

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108 views

### Cotorsion theory and its relative homology

Let (F(R), Cot(R)) be a cotorsion theory, Such that F(R) is the class of flat R-modules and Cot(R) the cotorsion modules. Why this is true that, For $ N\in Cot(R) $,
$ \text{Ext}_{F(R)}^i(M, N)\cong ...

**3**

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175 views

### (Co)limit computations for diagrams of Vector Spaces

Fix a field $K$ and consider a finite directed graph $\Gamma$ where multiple edges between a pair of vertices are allowed so long as the total number of edges is finite. Associate to each vertex $v$ a ...

**4**

votes

**2**answers

257 views

### Origin of notion of “split Grothendieck group”?

In the construction of Soergel's bimodules in representtion theory , it's essential for him to work with split Grothendieck groups. Here he starts with a certain small additive category ...

**0**

votes

**1**answer

372 views

### Are Verma modules universally characterised?

I'm having trouble understanding the definition of Verma Module in wikipedia. It later goes to show that it satisfies what appears to be a universal property (which I'm also having trouble ...

**4**

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240 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 ...

**3**

votes

**0**answers

196 views

### Comultiplication on Schur functors (& functions). Can it be seen from categorical perspective ?

Consider category of vector spaces. Consider functors from it to itself.
They actually form an algebra - since vector spaces can be added and tensor multiplied.
Question Is there co-product on this ...

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vote

**1**answer

143 views

### why the concept of compactly (or well) generated in triangulated categories is introduced?.

why the concept of compactly (or well) generated in triangulated categories is introduced?

**1**

vote

**4**answers

275 views

### Role of fiber functor monoidal structure in Tannakian bialgebra reconstruction

Shahn Majid presents in section 9.4 of "Foundatations of Quantum Group Theory" a Tannaka-type reconstruction theorem for producing a $k$-bialgebra out of its $k$-linear monoidal category of modules ...

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**0**answers

192 views

### Category of modules over a coPoisson-bialgebra

Fix a ground commutative ring $k$. A coPoisson-bialgebra is a bialgebra $H$ equipped with a linear mapping $\pi:H \rightarrow H \otimes_k H$ s.t.
$\pi$ is a coLie bracket
$\pi$ is a coderivation
...

**5**

votes

**1**answer

343 views

### Morita equivalence of acyclic categories

(Crossposted from math.SE.)
Call a category acyclic if only the identity morphisms are invertible and the endomorphism monoid of every object is trivial. Let $C, D$ be two finite acyclic categories. ...

**3**

votes

**2**answers

546 views

### Indecomposable projectives correspond to irreducibles - reference

Hello,
We have the following assertion:
In an abelian category that has enough projectives and in which every object has finite length, indecomposable projectives correspond bijectively to ...

**2**

votes

**0**answers

187 views

### localization functor for category of quasi coherent sheaves of $Fl_{\infty/2}$?

From series papers of Frenkel-Feign-Gaitsgory-Braveman and Beilinson Drinfeld and their school we know that the algebraic geometric definition and theory of semi infinite flag manifold were not ...

**2**

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**3**answers

275 views

### Can the 'linkages' between equivalent extensions of modules of an algebraic group be taken to have bounded length?

I have a conjecture concerning how "tightly" two equivalent $n$-fold extensions of modules over an algebraic group over a field might
be "linked". I suspect that the
question has been already ...

**7**

votes

**2**answers

509 views

### “Composition of Morita equivalences” or “Morita equivalence and the Nakayama functor”

This problem occured to me, when trying to find a Morita invariant for finite dimensional algebras.
Suppose $\Lambda$ and $\Gamma$ are two self-injective $k$-algebras ($k$ being a field) which are ...

**21**

votes

**1**answer

844 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 ...

**0**

votes

**1**answer

253 views

### Can injective resolutions be 'enlarged' (or shrunk) to admit only injective maps from extensions?

Let $M$ and $N$ be $R$-modules for some ring $R$. There is a standard result involving the computation of $\text{Ext}^n(M,N)$, using projective resolutions, which says that you can always choose a ...

**4**

votes

**2**answers

1k views

### Why SU(3) is not equal to SO(5)?

I am asking in the sense of isometry groups of a manifold. SU(3) is the group of isometries of CP2, and SO(5) is the group of isometries of the 4-sphere. Now, it happens that both manifolds are ...

**3**

votes

**3**answers

730 views

### Fractional Quantum Hall Effect - Mathematics

Just to include something that starts to answer my own question Topological Quantum Computation Lecture notes covers a lot of the Mathematics of the Fractional Quantum Hall effect, or topological ...

**6**

votes

**1**answer

707 views

### Conformal Field Theory and Langlands

I'm a Mathematics masters student currently
studying some aspects of TQFT. I'm interested in Langlands, mainly
because it sounds oppressive! Is anyone familiar with any links between
CFT and ...

**2**

votes

**2**answers

505 views

### endomorphism rings of indecomposable objects

Suppose $\mathcal C$ is a preadditive, Karoubi category with a zero object. What further assumptions on $\mathcal C$ are required to ensure that the endomorphism ring of an indecomposable object is a ...

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190 views

### Is there a common framework for Tannaka and Gabriel-Ulmer reconstruction theorems?

Gabriel-Ulmer duality is a biequivalence between the 2-category of finite limit categories and the 2-category of locally finitely presentable categories. It allows for the reconstruction of a theory ...

**17**

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**3**answers

1k views

### Why the BGG category O?

Given a finite-dimensional semisimple complex Lie algebra $\mathfrak{g}$, the Bernstein-Gelfand-Gelfand category $\mathcal O$ is the full subcategory of $\mathfrak g$-modules satisfying some ...

**6**

votes

**1**answer

1k views

### What is a Specht module?

I'm studying the structure of the Specht module for $S_n$ and I would like to know if there is some generalizations of this structure for Weyls groups or Coxeter groups.
Also, I'm interest to know ...

**14**

votes

**1**answer

437 views

### Is there a good notion of “induction” for representations of 2-categories?

One of the most important observations in the representation theory of algebras is that if one has a subalgebra $A\subset B$, then these is a functor $B\otimes_A -\colon A\operatorname{-pmod}\to ...

**9**

votes

**1**answer

488 views

### What's an example of a locally presentable category “in nature” that's not $\aleph_0$-locally presentable?

Recall the notion of locally presentable category (nLab): $\DeclareMathOperator{\Hom}{Hom}$
Definition: Fix a regular cardinal $\kappa$; a set is $\kappa$-small if its cardinality is strictly less ...

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votes

**2**answers

553 views

### Which monoidal categories are equivalent to their centers?

Let $\mathcal C$ be a monoidal category. Recall that the (Drinfel'd) center of $\mathcal C$ is the braided monoidal category $Z(\mathcal C)$ with:
Objects: pairs $M \in \mathcal C$ and $\mu: ...

**4**

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**2**answers

396 views

### Can a commutative, associative “multiplication” on an infinite-dimensional vector space be an isomorphism?

Let $V$ be a vector space (over $\mathbb C$, but I don't think it matters), and $m: V\otimes V \to V$ a "multiplication" that is associative and commutative (but I do not demand that it is unital). ...

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votes

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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 ...

**9**

votes

**1**answer

503 views

### Why not _co_free modules?

Let $R$ be a ring, and $R\text{-Mod}$ its category of all left modules. There is a "forgetful" functor $\operatorname{Forget}: R\text{-Mod} \to \text{AbGp}$, which is additive, continuous, and ...

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votes

**2**answers

426 views

### What are examples of cogenerators in R-mod?

Fill in the blank, please :)
Let $\mathcal C$ be a complete and cocomplete abelian category. A generator in $\mathcal C$ is an object $X \in \mathcal C$ such that every object $Y \in \mathcal C$ ...

**10**

votes

**1**answer

238 views

### What is known about higher-categorical reconstruction theorems? (reference request)

The answer to my question is almost certainly "not much" — at least, I've asked a few people, and that was their answer. But I'd like to refine this answer, and MathOverflow seems like the best ...

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votes

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659 views

### What makes the stable module category stable?

When geometrically flavoured words like "mapping cone" or "chain homotopy" crop up in homological algebra, there's usually a good reason. (In this case, looking at the chain complexes associated with ...

**5**

votes

**1**answer

490 views

### Looking for references talking about category of topological vector spaces

It's known that category of topological vector spaces is not abelian but quasi-abelian or exact category. I am looking for the references playing with this category(category theory). All the related ...

**10**

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**4**answers

564 views

### When are modules and representations not the same thing?

I've been trying for a while to get a real concrete handle on the relationship between representations and modules. To frame the question, I'll put here the standard situation I have in mind:
A ring ...

**8**

votes

**2**answers

396 views

### Applications of classifying thick subcategories

So, relatively recently, Balmer introduced this notion of a spectrum for a tensor triangulated category and used it to prove a generalization of a classification theorem done in several areas of ...

**10**

votes

**1**answer

1k views

### Why did people originally like Frobenius algebras?

These days, lots of people are excited by Frobenius algebras because commutative Frobenius algebras are the same thing as 2D topological quantum field theories.
...but this seems like teaching an old ...

**7**

votes

**1**answer

560 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
AndrĂ© Joyal ...

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vote

**2**answers

286 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 ...

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vote

**1**answer

462 views

### Is simple non-holonomic D-module a local concept?

It is well known that we can use the Riemann-Hilbert correspondence to describe holonomic D-modules in terms of a category of perverse sheaves on some variety $X$. And $U|\rightarrow Perv(U)$ is a ...

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votes

**3**answers

303 views

### What is an obviously coordinate-independent description of the Chevellay-Eilenberg complex for a Lie algebroid?

I've read in many places, including the n-Lab page, that a Lie algebroid (which I think of as in the first definition on the n-Lab page) is the same as a vector bundle $A \to X$ and a (properties?) ...

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votes

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638 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: ...