2.3. Representations 9

Definition 2.2.2. A unit in an associative algebra A is an element

1 ∈ A such that 1a = a1 = a.

Proposition 2.2.3. If a unit exists, it is unique.

Proof. Let 1, 1 be two units. Then 1 = 11 = 1 .

From now on, by an algebra A we will mean an associative algebra

with a unit. We will also assume that A = 0.

Example 2.2.4. Here are some examples of algebras over k:

1. A = k.

2. A = k[x1,...,xn] — the algebra of polynomials in variables

x1,...,xn.

3. A = EndV — the algebra of endomorphisms of a vector space

V over k (i.e., linear maps, or operators, from V to itself). The

multiplication is given by composition of operators.

4. The free algebra A = k

x1,...,xn . A basis of this algebra

consists of words in letters x1,...,xn, and multiplication in this basis

is simply the concatenation of words.

5. The group algebra A = k[G] of a group G. Its basis is

{ag,g ∈ G}, with multiplication law agah = agh.

Definition 2.2.5. An algebra A is commutative if ab = ba for all

a, b ∈ A.

For instance, in the above examples, A is commutative in cases 1

and 2 but not commutative in cases 3 (if dim V 1) and 4 (if n 1).

In case 5, A is commutative if and only if G is commutative.

Definition 2.2.6. A homomorphism of algebras f : A → B is a

linear map such that f(xy) = f(x)f(y) for all x, y ∈ A and f(1) = 1.

2.3. Representations

Definition 2.3.1. A representation of an algebra A (also called a

left A-module) is a vector space V together with a homomorphism

of algebras ρ : A → EndV .