Math 662 Quiver Representations Homework Assignment 3
Due Friday March 7
Let k be an algebraically closed field. All modules are right modules.
1. Let Q be the quiver
α
•R
β
Let n be a positive integer, and let M = k n be the representation of Q for
which




0 1 0 ··· 0
0
.. 

 1 0


0 1
. 




...


.


.
0 1
φα = 
.
.
0  and φβ = 
 .



.
.
..
 ..

.. 1 

0 ··· 0 1 0
0
Show that M is simple.
2. Let Q be the quiver
α
1
* 2
•
•j
β
Show that kQ has infinitely many simple modules.
3. Let Q be the quiver
2
β
4
•`
•_
α

•1
δ
3•
γ
~
Describe all simple kQ-modules and all projective indecomposable kQmodules (by giving a nice vector space basis of each).
4. Let Q be the Kronecker quiver
α
1
•
5
* 2
•
β
For each λ ∈ k, let Hλ be the representation of Q for which (Hλ )1 = k,
(Hλ )2 = k, φα = id, and φβ = λid, where id is the identity map. Show
that, for each λ, Hλ is indecomposable, and that Hλ ∼
= Hµ if and only if
λ = µ.