Tagged Questions

A quiver is an oriented graph which might contain multiple edges and loops. The terminology is used in representation-theory of finite dimensional algebras, where one considers functors from this graph, viewed as a category, to the category of vector spaces.

2k 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, ...
575 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", ...
95 views

When are infinite dimensional path algebras hereditary

The title says mostly everything. Suppose we have a quiver, maybe with relations and cycles. Is it known when the path algebra modulo relations is hereditary? Especially in the case that the path ...
191 views

44 views

Quiver algebra as a wreath product?

I'm having trouble understanding a definition of a quiver Hecke algebra. Suppose $k$ is a commutative ring, and $\Omega$ a finite set. We build a quiver $Q_{\Omega,n}$ with vertex set $\Omega^n$. ...
57 views

What can we do about the indecomposable representations of wild quivers and wild algebras?

I know that using finite number of parameters we can not describe indecomposable modules of wild quivers, but is it possible for us to describe them using infinitely many parameters for at least some ...
53 views

79 views

Endomorphism ring of indecomposable representations

Let $Q$ be the quiver given by an $n\times n$ grid where every square commutes and let $F:Q\to {\rm vec_k}$ be an indecomposable (finitely dimensional) representation of $Q$. I am interested in ...
231 views

How to compute the ordinary quiver of $B = \operatorname{End}_A(T_{A})$?

I am reading the book Elements of the Representation Theory of Associative Algebras: Volume 1: Techniques of Representation Theory by Ibrahim Assem, Daniel Simson, Andrzej Skowronski. Let $A$ be a $K$...
55 views

The sum of trivial paths for a finite quiver is 1?

let $Q=(E_0, E_1)$ be a quiver and let $P_Q$ be a path algebra of $Q$. Let $p_i$ be the trivial path associated to each vertex $i$ in $E_0$. Then why is $\sum_{i\in E_0} p_i=1$ for a finite quiver? ...
21 views

How to define a quiver of basic and non-connected associative algebras

Recently, I am reading a book "Elements of the Representation Theory of Associative Algebras". Let $A$ be a basic and connected finite dimensional $~\mathbb{K}$-algebra and $\{e_1, e_2, \cdots, e_n\}$...
57 views

Projective Indecomposable modules of quiver algebra

I am trying to prove that the module $kQe_i$ of paths starting from a node $i$ in an acyclic finite quiver $Q$ is a projective indecomposable module for $kQ$. The projectivity is clear since I can ...
47 views

The indecomposable projective A-modules

Let Q be the quiver bound by $αβ = 0$, $γδ = 0$. The indecomposable projective A-modules are given by where $A=KQ/I$. This an example in Assem-Simson-Skowronski book (Elements of the ...
59 views

Computing quotient representations and Hom set fort wo representations

Consider the representation $M$ defined by We want to find all subrepresentations quotient representations of $M$, and $\mathrm{Hom}(M,N)$, where $N$ is a representation with $N \cong M$. I put B ...
Consider a tame quiver $Q$ whose underlying undirected graph is connected. So that undirected graph is one of the extended, simply laced Dynkin graphs; it's either $\tilde A_n$ for some $n\ge 1$ or $\... 1answer 78 views Question about the connectivity of the ordinary quiver of a connected algebra. I am reading the book Elements of Representation Theory of Associative Algebras, Volume 1. I have some questions about the connectivity of the ordinary quiver of a connected algebra. On page 61, ... 1answer 85 views finite dimensional algebras without bound quiver representation (1) For the$\mathbb{R}$-algebra$\mathbb{C}$, ($\mathbb{R}$real number field,$\mathbb{C}$complex number field) there is no quiver$Q $such that$\mathbb{C}\cong \mathbb RQ/\mathcal{I}$with$\...
Let $Q$ be a finite quiver and denote the stationary parts of $Q$ by $e_{i}$. Suppose we have two arrows $f,g$ such that their composition $f \circ g$ is equal to $e_{i}$. Does this always implies ...