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 .
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