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 =

n≥0

A[n],

and A[n] · A[m] ⊂ A[n + m]. If A[n] is finite dimensional, it is useful

to consider the Hilbert series hA(t) =

∑

dim

A[n]tn

(the generating

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 +

t2

+ · · · +

tn

+ · · · =

1

1 − t

.

Find the Hilbert series of the following graded algebras:

(a) A = k[x1,...,xm] (where the grading is by degree of polyno-

mials).

(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.