2.3. Representations 11

linear operator which commutes with the action of A, i.e., φ(av) =

aφ(v) for any v ∈ V1. A homomorphism φ is said to be an isomor-

phism of representations if it is an isomorphism of vector spaces.

The set (space) of all homomorphisms of representations V1 → V2 is

denoted by HomA(V1,V2).

Note that if a linear operator φ : V1 → V2 is an isomorphism of

representations, then so is the linear operator

φ−1

: V2 → V1 (check

it!).

Two representations between which there exists an isomorphism

are said to be isomorphic. For practical purposes, two isomorphic

representations may be regarded as “the same”, although there could

be subtleties related to the fact that an isomorphism between two

representations, when it exists, is not unique.

Definition 2.3.7. Let V1,V2 be representations of an algebra A.

Then the space V1 ⊕ V2 has an obvious structure of a representation

of A, given by a(v1 ⊕ v2) = av1 ⊕ av2. This representation is called

the direct sum of V1 and V2.

Definition 2.3.8. A nonzero representation V of an algebra A is said

to be indecomposable if it is not isomorphic to a direct sum of two

nonzero representations.

It is obvious that an irreducible representation is indecomposable.

On the other hand, we will see below that the converse statement is

false in general.

One of the main problems of representation theory is to classify

irreducible and indecomposable representations of a given algebra up

to isomorphism. This problem is usually hard and often can be solved

only partially (say, for finite dimensional representations). Below we

will see a number of examples in which this problem is partially or

fully solved for specific algebras.

We will now prove our first result — Schur’s lemma. Although

it is very easy to prove, it is fundamental in the whole subject of

representation theory.

Proposition 2.3.9 (Schur’s lemma). Let V1,V2 be representations of

an algebra A over any field F (which need not be algebraically closed).