0
votes
1answer
39 views

In a semi-simple module, any submodule is a direct factor?

I need help understanding the following : In a semi-simple module, any submodule is a direct factor (this is sometimes taken as the definition of semi-simple) (i) How is this equivalent to the ...
4
votes
0answers
52 views

Left and Right minimal homomorphisms.

In the literature on representation theory of finite dimensional algebras, a left (and similarly right) minimal homomorphism is defined as the following: For a pair of modules $L $ and $M$ in ...
4
votes
0answers
22 views

“twisted” powers in symmetric monoidal categories

Suppose $C$ is a symmetric monoidal category with monoidal product $\wedge$, $X$ is a $G$-object for some finite group $G$ (say), and $T$ is a finite $G$-set of size $n$. The $n$-fold monoidal power ...
4
votes
0answers
47 views

~ The important use of Frobenius–Schur indicators and Frobenius-Schur exponents ~

I had asked a question on the uses of conjugacy class and centralizer. I had receive various helpful feedback. I appreciate them. Here I like to get some feedback on the Frobenius–Schur indicator. ...
1
vote
0answers
32 views

Does the representation ring functor preserve limits?

If I have a diagram of groups $\{H_J\}$ and let $G$ be the limit of that diagram, how well does the representation ring functor "preserve the limit", IE: If I have $\lim_{J} H_J = G$ is it true that ...
3
votes
2answers
54 views

Isomorphism $\text {Rep}_{G,k}\cong \space \text {Mod}_{k[G]} $

Let $G$ be a group and $k$ a field. Proposition: There is an isomorphism of categories $F:\text {Rep}_{G,k}\rightarrow \space \text {Mod}_{k[G]} $. I begun by proving that there is a functor ...
9
votes
1answer
126 views

Category of Lie group representations equivalent to the category of representations of their Lie algebra

Let $G$ be a lie group and $\mathfrak{g}$ its lie algebra. Consider the category $Rep(G)$ of finite dimensional representations of $G$ and the category $Rep(\mathfrak{g})$ of finite dimensional ...
3
votes
1answer
86 views

How do you show Grothendieck group is free abelian on simple objects?

Given an abelian category C of finite length (i.e., every object has a Jordan-Holder filtration), let $A$ be the set of isomorphism classes of simple objects. there's an obvious map $\mathbb Z^A \to ...
3
votes
0answers
51 views

Where does a modular tensor category come from?

I have studied the definition of a modular tensor category. I jumped into this subject and almost have no background. My question is: what kind of mathematics does a modular tensor category ...
10
votes
2answers
304 views

Path Algebra for Categories

For a while I had been thinking that the path algebra of a quiver $Q$ over a commutative ring $R$ is the same as the "category ring" $R[P]$ (analogous to "group ring", "monoid ring", "semigroup ring", ...
7
votes
2answers
577 views

Why are (representations of ) quivers such a big deal?

Quivers are directed graphs where loops and multi-arrows are allowed. And we can talk about representations of quivers by assigning each vertex a vector space and each arrow a homomorphism. Moreover, ...
1
vote
0answers
33 views

Showing that representations are isomorphic if the “forgetful representations” are isomorphic.

Suppose $V_{\pi_1}$ and $V_{\pi_2}$ are irreducible admissible smooth representations of $G$. For $K$ an open compact subgroup of $G$, let $V_\pi^K$ be the set of $v$ in $V_\pi$ such that $\pi(k)v=v$. ...
1
vote
0answers
62 views

2-morphisms from spans of spans

I have a question about the construction of 2-morphisms from spans of spans in the paper "2-vector spaces and groupoid" by Jeffrey Morton . Suppose we have a span of span of groupoids as follows and ...
3
votes
1answer
255 views

Left adjoint and right adjoint/ Nakayama isomorphism

I am reading a paper "2-vector spaces and groupoid" by Jeffrey Morton and I need a help to understand the following. Let $X$ and $Y$ be finite groupoids. Let $[X, \mathbb{Vect}]$ be a functor ...
4
votes
1answer
67 views

Associativity, Jacobi, and self-action representations

About a year and a half ago, I was at a talk by Martin Hyland where he suggested that the Jacobi identity is to the associative law as the anticommutative law is to the commutative law. I think this ...
3
votes
1answer
236 views

What is the categorical perspective on representations of topological groups?

One categorical definition of a group $G$ is that it is a category $C$ with a single object $X$ such that every morphism in the set $C(X,X)$ is invertible, i.e. such that $C(X,X)$ is precisely the ...
3
votes
2answers
191 views

Morita equivalence of acyclic categories

(Crossposted to MathOverflow.) 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 ...
1
vote
1answer
89 views

Equivalent module categories

Let $A$ and $B$ be rings and let $A\text{-mod}$ and $B\text{-mod}$ be their abelian module categories. Let $F:A\text{-mod}\to M\text{-mod}$ and $F':B\text{-mod}\to A\text{-mod}$ be functors which ...
2
votes
1answer
179 views

Categorical definition of permutation representation

In Mac Lane's, Categories for the Working Mathematician, on p.15 ex.3c) it asks to interpret "functor" when F: (group G)-->Set is a permutation representation of G. Here is where I get stuck, G is ...
4
votes
1answer
207 views

Reference request: Deligne's reconstruction theorem

I've heard this result referenced a few times on MO now. It is supposed to be a theorem of Deligne that gives some natural conditions under which an (abelian?) tensor category $C$ is the category of ...
18
votes
2answers
583 views

Categorical description of algebraic structures

There is a well-known description of a group as "a category with one object in which all morphisms are invertible." As I understand it, the Yoneda Lemma applied to such a category is simply a ...
5
votes
0answers
96 views

How to prove “The maps factoring through an injective object are precisely the null-homotopic maps”

Thanks for your attention, I'm an undergraduate. I'm reading the book of Dieter Happel, Triangulated categories in the representation theory of finite dimensional algebras, I cannot prove this ...