Math 662 Quiver Representations Homework Assignment 1
Due Friday January 31
Let k be an algebraically closed field.
1. Let Q be the quiver
β
/•
/•
•
(a) Show that the path algebra kQ is isomorphic to the algebra of upper
triangular 3 × 3 matrices by exhibiting an explicit isomorphism.
(b) Find rad(kQ).
α
2. More generally, let Q be the quiver
•
α1
/
α2
•
/
αn−1
•···•
/
•
Outline a proof that the path algebra kQ is isomorphic to the algebra of
upper triangular n × n matrices. (You need not check details.)
3. Let Q be the quiver
α
(
•h
•
β
Show that kQ contains a subalgebra isomorphic to kha, bi/(a2 , b2 ).
4. Let Q be the quiver
α
6
•
(
•
β
Give a basis for kQ as a vector space, and a multiplication table. (This is
the Kronecker quiver, and kQ is the Kronecker algebra.)