2.8. Quivers 21
Definition 2.8.8. A subrepresentation of a representation (Vi,xh)
of a quiver Q is a representation (Wi,xh) where Wi ⊆ Vi for all i ∈ I
and where xh(Wh ) ⊆ Wh and xh = xh|Wh : Wh −→ Wh for all
h ∈ E.
Definition 2.8.9. The direct sum of two representations (Vi,xh)
and (Wi,yh) is the representation (Vi ⊕ Wi,xh ⊕ yh).
As with representations of algebras, a nonzero representation (Vi)
of a quiver Q is said to be irreducible if its only subrepresentations
are (0) and (Vi) itself, and it is said to be indecomposable if it is not
isomorphic to a direct sum of two nonzero representations.
Definition 2.8.10. Let (Vi,xh) and (Wi,yh) be representations of
the quiver Q. A homomorphism ϕ : (Vi) −→ (Wi) of quiver
representations is a collection of maps ϕi : Vi −→ Wi such that
yh ◦ ϕh = ϕh ◦ xh for all h ∈ E.
Problem 2.8.11. Let A be a Z+-graded algebra, i.e., A =
and A[n] · A[m] ⊂ A[n + m]. If A[n] is finite dimensional, it is useful
to consider the Hilbert series hA(t) =
function of dimensions of A[n]). Often this series converges to a ratio-
nal function, and the answer is written in the form of such a function.
For example, if A = k[x] and deg(xn) = n, then
hA(t) = 1 + t +
+ · · · +
+ · · · =
1 − t
Find the Hilbert series of the following graded algebras:
(a) A = k[x1,...,xm] (where the grading is by degree of polyno-
(b) A = k x1,...,xm (the grading is by length of words).
(c) A is the exterior (= Grassmann) algebra ∧k[x1,...,xm] gen-
erated over some field k by x1,...,xm with the defining relations
xixj + xjxi = 0 and xi 2 = 0 for all i, j (the grading is by degree).
(d) A is the path algebra PQ of a quiver Q (the grading is defined
by deg(pi) = 0, deg(ah) = 1).
Hint: The closed answer is written in terms of the adjacency
matrix MQ of Q.