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

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