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
: V2 → V1 (check
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
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
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).