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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