2.3. Representations 13

every subspace of V is a subrepresentation. But V is irreducible, so

0 and V are the only subspaces of V . This means that dim V = 1

(since V = 0).

Example 2.3.14. 1. A = k. Since representations of A are simply

vector spaces, V = A is the only irreducible and the only indecom-

posable representation.

2. A = k[x]. Since this algebra is commutative, the irreducible

representations of A are its 1-dimensional representations. As we

discussed above, they are defined by a single operator ρ(x). In the 1-

dimensional case, this is just a number from k. So all the irreducible

representations of A are Vλ = k, λ ∈ k, in which the action of A

is defined by ρ(x) = λ. Clearly, these representations are pairwise

nonisomorphic.

The classification of indecomposable representations of k[x] is

more interesting. To obtain it, recall that any linear operator on

a finite dimensional vector space V can be brought to Jordan nor-

mal form. More specifically, recall that the Jordan block Jλ,n is the

operator on

kn

which in the standard basis is given by the formulas

Jλ,nei = λei + ei−1 for i 1 and Jλ,ne1 = λe1. Then for any linear

operator B : V → V there exists a basis of V such that the matrix

of B in this basis is a direct sum of Jordan blocks. This implies that

all the indecomposable representations of A are Vλ,n =

kn,

λ ∈ k,

with ρ(x) = Jλ,n. The fact that these representations are indecom-

posable and pairwise nonisomorphic follows from the Jordan normal

form theorem (which in particular says that the Jordan normal form

of an operator is unique up to permutation of blocks).

This example shows that an indecomposable representation of an

algebra need not be irreducible.

3. The group algebra A = k[G], where G is a group. A represen-

tation of A is the same thing as a representation of G, i.e., a vector

space V together with a group homomorphism ρ : G → Aut(V ),

where Aut(V ) = GL(V ) denotes the group of invertible linear maps

from the space V to itself (the general linear group of V ).