10 2. Basic notions of representation theory

Similarly, a right A-module is a space V equipped with an

antihomomorphism ρ : A → EndV ; i.e., ρ satisfies ρ(ab) = ρ(b)ρ(a)

and ρ(1) = 1.

The usual abbreviated notation for ρ(a)v is av for a left mod-

ule and va for a right module. Then the property that ρ is an

(anti)homomorphism can be written as a kind of associativity law:

(ab)v = a(bv) for left modules, and (va)b = v(ab) for right modules.

Remark 2.3.2. Let M be a left module over a commutative ring

A. Then one can regard M as a right A-module, with ma := am.

Similarly, any right A-module can be regarded as a left A-module. For

this reason, for commutative rings one does not distinguish between

left and right A-modules and just calls them A-modules.

Here are some examples of representations.

Example 2.3.3. 1. V = 0.

2. V = A, and ρ : A → EndA is defined as follows: ρ(a) is the

operator of left multiplication by a, so that ρ(a)b = ab (the usual

product). This representation is called the regular representation of

A. Similarly, one can equip A with a structure of a right A-module

by setting ρ(a)b := ba.

3. A = k. Then a representation of A is simply a vector space

over k.

4. A = k x1,...,xn . Then a representation of A is just a vec-

tor space V over k with a collection of arbitrary linear operators

ρ(x1),...,ρ(xn) : V → V (explain why!).

Definition 2.3.4. A subrepresentation of a representation V of

an algebra A is a subspace W ⊂ V which is invariant under all the

operators ρ(a) : V → V , a ∈ A.

For instance, 0 and V are always subrepresentations.

Definition 2.3.5. A representation V = 0 of A is irreducible (or

simple) if the only subrepresentations of V are 0 and V .

Definition 2.3.6. Let V1,V2 be two representations of an algebra A.

A homomorphism (or intertwining operator) φ : V1 → V2 is a