1
vote
1answer
28 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 ...
1
vote
1answer
40 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 ...
1
vote
1answer
39 views

Computing Path Algebra of a Quiver

Let $Q$ be a quiver over defined as follows Then $KQ\cong$ $\begin{pmatrix}K&K&K\\0&K&K\\0&0&K\end{pmatrix}$, where $KQ$ is just the path algebra. What the professor did was ...
2
votes
1answer
48 views

isomorphism classes of representations of a quiver

Classify all isomorphism classes of representations of dimension vector 1 and 2 of the following quiver The professor briefly did the solution, but I could not understand what was going on. What he ...
1
vote
1answer
44 views

Question for recommending a good textbook in representation of quivers

I am taking representation of quivers, and the lecture notes seems not enough. So could you recommend a good textbook for this course. There is a new book "Quiver Representations, by Ralf Schiffler" ...
2
votes
1answer
58 views

Indecomposable quiver representations

Is there are any way to found indecomposable representation of a given quiver explicitely if it's dimention vector is given?
4
votes
1answer
99 views

When are path algebras of quivers hereditary.

Suppose we have a finite quiver with relations, possibly with oriented cycles. Is it known when the path algebra of this quiver (with relations) is hereditary?
2
votes
1answer
68 views

A finite-type quiver has no self-loops

I am reading through Etingof et al's notes on representation theory, and they assert in Exercise 5.4(c) on page 80 that a finite-type quiver has no self-loops. I think the way to show this is to ...
10
votes
2answers
291 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", ...
3
votes
1answer
182 views

Projective indecomposables versus general indecomposables

Given a finite dimensional algebra, what is the exact relation between the indecomposable projective modules, and a general indecomposable module? In the case of an oriented quiver without cycles for ...
2
votes
1answer
106 views

Cartan or Coxeter matrix of an algebra of infinite global dimension

Let $(Q, I)$ be a bound quiver such that $A=KQ/I$ has infinite global dimension. I want to ask the following questionss: (1) Is the Cartan matrix $C_A$ of $A$ invertible in the matrix ring ...
4
votes
1answer
174 views

how to find all simple modules for the given path algebra

Let $A = KQ$, where $Q$ is the quiver $$\begin{array}{ccc} & \alpha & \\ 1 & \rightleftarrows & 2 \\ & \beta& \end{array}$$ are there simple right $A$-modules with dimension ...
6
votes
1answer
219 views

how to get the injective envelope and projective cover of a given module

Given a bound quiver $(Q, I)$ and a representation $M$ of $Q$, how to get the injective envelope and projective cover of $M$? how to give the corresponding essential monomorphism and superfluous ...
1
vote
1answer
81 views

Stationary paths

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 ...
2
votes
1answer
245 views

The projective module in the quiver representation

I study the quiver just some weeks and I cannot understand the projective module in the quiver representation well.Here are some questions: Suppose $Q$ is a quiver, $a\in Q_{0}$. 1)Show that the ...