The quiver tag has no wiki summary.
10
votes
2answers
115 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
95 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
162 views
Is every finitely dimensional basic $K$-algebra isomorphic to a bound quiver algebra?
Given a positive integer $n$, how to classify $n$-dimensional basic $K$-algebras?, where $K$ is algebraically closed.
For $n=3$, Let $A=\left[
\begin{array}{ccc}
...
2
votes
1answer
60 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 ...
3
votes
1answer
85 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
86 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 ...
2
votes
1answer
87 views
Given a quiver, how do you get the indecomposable injective modules from indecomposable projective modules?
Given a quiver, we know that it is easy to get the indecomposable projective modules, but the indecomposable injective modules are not easy to get.
How do you get the indecomposable injective ...
3
votes
1answer
167 views
admissible ideals
How to prove the following conclusion :
For any finite quiver $Q$, an ideal $I$ of $KQ$, contained
in $R^2_Q$, is admissible if and only if, for each cycle
$\sigma$ in $Q$, there exists $s \geq 1$ ...
1
vote
1answer
58 views
Is an abstract simplicial complex a quiver?
Let $\Delta$ be an abstract simplicial complex. Then for $B\in \Delta$ and $A\subseteq B$ we have that $A\in\Delta$. If we define $V$ to be the set of faces of $\Delta$, construct a directed edge from ...
1
vote
1answer
31 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? ...
1
vote
1answer
66 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
155 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 ...
