Definitions and Examples

4.1 Representations of Associative Algebras

4.1.1 Definitions and Examples

We know from the previous chapter that every regular representation (ρ,V)of a reductive linear algebraic groupGdecomposes into a direct sum of irreducible rep- resentations (in particular, this is true whenGis a classical group). The same is true for finite-dimensional representations of a semisimple Lie algebrag. The next task is to determine the extent of uniqueness of such a decomposition and to find explicit projection operators onto irreducible subspaces ofV. In the tradition of modern mathematics we will attack these problems by putting them in a more general (ab- stract) context, which we have already employed, for example, in the proof of the theorem of the highest weight in Section 3.2.1.

R. Goodman, N.R. Wallach, Symmetry, Representations, and Invariants 175 Graduate Texts in Mathematics 255, DOI 10.1007/978-0-387-79852-3_

© Roe Goodman and Nolan R. Wallach 2009

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Definition 4.1.1.Anassociative algebraover the complex fieldCis a vector space AoverCtogether with a bilinear multiplication map

µ:A×A //A, x,y7→xy=µ(x,y),

such that(xy)z=x(yz). The algebraAis said to have aunit elementif there exists e∈Asuch thatae=ea=afor alla∈A. IfAhas a unit element it is unique and it will usually be denoted by 1.

Examples

1. LetV be a vector space over C (possibly infinite-dimensional), and let A= End(V)be the space ofC-linear transformations onV. ThenAis an associative al- gebra with multiplication the composition of transformations. When dimV=n<∞, then this algebra has a basis consisting of theelementary matrices e_{i j}that multiply by e_{i j}e_km=δ_jke_im for 1≤i,j≤n. This algebra will play a fundamental role in our study of associative algebras and their representations.

2.LetGbe a group. We define an associative algebraA[G], called thegroup algebra of G, as follows: As a vector space,A[G]is the set of all functions f :G //C such that thesupportof f (the set where f(g)6=0) is finite. This space has a basis consisting of the functions{δg:g∈G}, where

δ_g(x) =

1 ifx=g, 0 otherwise.

Thus an elementxofA[G]has a unique expression as a formal sum∑g∈Gx(g)δg

with only a finite number of coefficientsx(g)6=0.

We identify g∈G with the elementδg∈A[G], and we define multiplication on A[G] as the bilinear extension of group multiplication. Thus, given functions x,y∈A[G], we define their productx∗yby

∑g∈Gx(g)δg

∗ ∑h∈Gy(h)δh

=∑g,h∈Gx(g)y(h)δgh,

with the sum overg,h∈G. (We indicate the multiplication by∗ so it will not be confused with the pointwise multiplication of functions onG.) This product is as- sociative by the associativity of group multiplication. The identity elemente∈G becomes the unit elementδ_einA[G]andGis a subgroup of the group of invertible elements ofA[G]. The functionx∗yis called theconvolutionof the functionsxand y; from the definition it is clear that

(x∗y)(g) =∑hk=gx(h)y(k) =∑h∈Gx(h)y(h⁻¹g).

Ifϕ:G //His a group homomorphism, then we can extendϕ uniquely to a linear mapϕe:A[G] //A[H]by the rule

4.1 Representations of Associative Algebras 177

ϕe ∑_g∈Gx(g)δ_g=∑_g∈Gx(g)δ_ϕ(g).

From the definition of multiplication inA[G]we see that the extended mapϕeis an associative algebra homomorphism. Furthermore, ifψ:H //Kis another group homomorphism, thenψ]◦ϕ=ψe◦ϕ.e

An important special case occurs whenGis a subgroup ofHandϕis the inclu- sion map. Thenϕeis injective (since{δg}is a basis ofA[G]). Thus we can identify A[G]with the subalgebra ofA[H]consisting of functions supported onG.

3.Letgbe a Lie algebra overA. Just as in the case of group algebras, there is an associative algebraU(g)(the universal enveloping algebraofg) and an injective linear map j:g //U(g)such that j(g)generatesU(g)and

j([X,Y]) = j(X)j(Y)−j(Y)j(X)

(the multiplication on the right is inU(g); see Appendix C.2.1 and Theorem C.2.2).

SinceU(g)is uniquely determined byg, up to isomorphism, we will identifygwith j(g). Ifh⊂gis a Lie subalgebra then the Poincar´e–Birkhoff–Witt Theorem C.2.2 allows us to identifyU(h)with the associative subalgebra ofU(g)generated byh, so we have the same situation as for the group algebra of a subgroupH⊂G.

Definition 4.1.2.LetAbe an associative algebra overC. ArepresentationofAis a pair(ρ,V), whereVis a vector space overCandρ:A //End(V)is an associative algebra homomorphism. IfAhas an identity element 1, then we require thatρ(1) act as the identity transformationIVonV.

When the mapρis understood from the context, we shall callV anA-moduleand write avfor ρ(a)v. IfV,W are both A-modules, then we make the vector space V⊕W into anA-module by the actiona·(v⊕w) =av⊕aw.

IfU⊂Vis a linear subspace such thatρ(a)U⊂Ufor alla∈A, then we say that U isinvariantunder the representation. In this case we can define a representation (ρU,U)by the restriction ofρ(A)toU and a representation(ρ_V/U,V/U)by the natural quotient action of ρ(A)onV/U. A representation (ρ,V)isirreducibleif the only invariant subspaces are{0}andV.

Define Ker(ρ) ={x∈A: ρ(x) =0}. This is a two-sided ideal in A, and V is a module for the quotient algebra A/Ker(ρ) via the natural quotient map. A representationρisfaithfulif Ker(ρ) =0.

Definition 4.1.3.Let(ρ,V)and(τ,W)be representations ofA, and let Hom(V,W) be the space ofC-linear maps fromV toW. We denote by Hom_A(V,W)the set of allT∈Hom(V,W)such thatTρ(a) =τ(a)T for alla∈A. Such a map is called an intertwining operatorbetween the two representations or amodule homomorphism.

IfU ⊂V is an invariant subspace, then the inclusion mapU //V and the quotient mapV //V/U are intertwining operators. We say that the represen- tations (ρ,V) and(τ,W) are equivalent if there exists an invertible operator in Hom_A(V,W). In this case we write(ρ,V)∼= (τ,W).

The composition of two intertwining operators, when defined, is again an inter- twining operator. In particular, whenV =W andρ =τ, then Hom_A(V,V)is an associative algebra, which we denote by End_A(V).

Examples

1. Let A=C[x] be the polynomial ring in one indeterminate. LetV be a finite- dimensional vector space, and letT ∈End(V). Define a representation(ρ,V)ofA byρ(f) =f(T)forf∈C[x]. Then Ker(ρ)is the ideal inAgenerated by theminimal polynomialofT. The problem of finding a canonical form for this representation is the same as finding the Jordan canonical form forT (see Section B.1.2).

2. LetG be a group and letA=A[G]be the group algebra ofG. If (ρ,V) is a representation ofA, then the mapg7→ρ(δg)is a group homomorphism fromGto GL(V). Conversely, every representationρ:G //GL(V)extends uniquely to a representationρofA[G]onV by

ρ(f) =∑g∈G f(g)ρ(g)

for f ∈A[G]. We shall use the same symbol to denote a representation of a group and its group algebra.

SupposeW ⊂V is a linear subspace. If W is invariant under G andw∈W, thenρ(f)w∈W, sinceρ(g)w∈W. Conversely, ifρ(f)W ⊂W for all f ∈A[G], thenρ(G)W ⊂W, since we can take f =δ_gwithgarbitrary inG. Furthermore, an operatorR∈End(V)commutes with the action ofGif and only if it commutes with ρ(f)for all f∈A[G].

Two important new constructions are possible in the case of group representa- tions (we already encountered them in Section 1.5.1 whenGis a linear algebraic group). The first is thecontragredientordualrepresentation(ρ^∗,V^∗), where

hρ^∗(g)f,vi=hf,ρ(g⁻¹)vi

forg∈G,v∈V, and f ∈V^∗. The second is thetensor product(ρ⊗σ,V⊗W)of two representations defined by

(ρ⊗σ)(g)(v⊗w) =ρ(g)v⊗σ(g)w.

For example, let (ρ,V) and (σ,W) be finite-dimensional representations of G.

There is a representation π of G on Hom(V,W) by π(g)T =σ(g)Tρ(g)⁻¹ for T ∈Hom(V,W). There is a natural linear isomorphism

Hom(V,W)∼=W⊗V^∗ (4.1)

(see Section B.2.2). Here a tensor of the formw⊗v^∗gives the linear transforma- tionT v=hv^∗,viwfromV toW. Since the tensorσ(g)w⊗ρ^∗(g)v^∗gives the linear transformation

4.1 Representations of Associative Algebras 179

v7→ hρ^∗(g)v^∗,viσ(g)w=hv^∗,ρ(g)⁻¹viσ(g)w=σ(g)Tρ(g)⁻¹v,

we see thatπ is equivalent toσ⊗ρ^∗. In particular, the space Hom_G(V,W)ofG- intertwining maps betweenVandW corresponds to the space(W⊗V^∗)^GofG-fixed elements inW⊗V^∗.

We can iterate the tensor product construction to obtainG-modules^NkV=V^⊗k (thek-fold tensor product ofV with itself) withg∈Gacting by

ρ^⊗k(g)(v1⊗ ··· ⊗v_k) =ρ(g)v1⊗ ··· ⊗ρ(g)vk

on decomposable tensors. The subspaces S^k(V) (symmetric tensors) and ^VkV (skew-symmetric tensors) are G-invariant (see Sections B.2.3 and B.2.4). These modules are called thesymmetricandskew-symmetric powers ofρ.

The contragredient and tensor product constructions for group representations are associated with the inversion map g7→g⁻¹and thediagonal map g7→(g,g).

The properties of these maps can be described axiomatically using the notion of a Hopf algebra(see Exercises 4.1.8).

3.Letgbe a Lie algebra overC, and let(ρ,V)be a representation ofg. The univer- sal mapping property implies thatρ extends uniquely to a representation ofU(g) (see Section C.2.1) and that every representation ofgcomes from a unique repre- sentation ofU(g), just as in the case of group algebras. In this case we define the dualrepresentation(ρ^∗,V^∗)by

hρ^∗(X)f,vi=−hf,ρ(X)vi forX∈gand f ∈V^∗.

We can also define thetensor product(ρ⊗σ,V⊗W)of two representations by lettingX∈gact by

X·(v⊗w) =ρ(X)v⊗w+v⊗σ(X)w.

Whengis the Lie algebra of a linear algebraic groupGandρ,σare the differentials of regular representations ofG, then this action ofgis the differential of the tensor product of theGrepresentations (see Sections 1.5.2).

These constructions are associated with the mapsX7→ −XandX7→X⊗I+I⊗ X. As in the case of group algebras, the properties of these maps can be described axiomatically using the notion of aHopf algebra(see Exercises 4.1.8). Thek-fold tensor powers ofρ and the symmetric and skew-symmetric powers are defined by analogy with the case of group representations. HereX∈gacts by

ρ^⊗k(X)(v₁⊗ ··· ⊗v_k) =ρ(X)v₁⊗ ··· ⊗v_k+v₁⊗ρ(X)v₂⊗ ··· ⊗v_k +···+v₁⊗ ··· ⊗ρ(X)vk

on decomposable tensors. This action extends linearly to all tensors.