In group theory, a linear representation of a groupGis a homomorphismρ:G→GLV(k) for some vector spaceV overk. To see what should be the equivalent notion for affine groups, we shall prove a proposition aboutR-linear endomorphisms of V ⊗kR. for R∈Algk.
Proposition 5.1.1. For ak-moduleV, everyR-linear endomorphismϕ:V⊗kR→V⊗kR
is determined by its restriction to V.
Proof. Recall thatV ⊗kR is an R-module with multiplication defined byr(v⊗s) =v⊗rs expanded linearly. So givenV →V ⊗kR, we have that each v⊗r =r(v⊗1)∈V ⊗R. By
R-linearity the image of v⊗r is thus determined by v⊗1.
It follows that GLV is a group functor: For eachR∈Algk, GLV(R) = AutR(V⊗R) is a group with group multiplication defined by composition. Since each endomorphism (overR)
V⊗R→V⊗Ris defined by its restriction toV →V⊗R, for each homomorphismϕ:R→S
we define GLV(ϕ) : AutR(V ⊗R) → AutS(V ⊗S) by sending each f ∈ AutR(V ⊗R) to
g∈AutS(V ⊗S) defined byg|V = (id⊗ϕ)◦f|V :V →V ⊗S. It is clear that GLV(ϕ) is a group homomorphism by definition. Thus, we define alinear representation onV of an affine group G to be a homomorphism Φ : G → GLV. As one would expect, a subrepresentation
of V is a linear subspace W ⊆V such that for Φ(R)(g)(W) =W over all R-points and all
g∈ G(R).
Let G = Spec(A) be an affine group. As discussed in Chapter 1, Yoneda’s lemma tells us that the set of natural transformations (not group homomorphisms) Φ : G → GLV is in bijection with GLV(A) and the bijection is given by the map Φ 7→ Φ(A)(idA). Since eachf ∈GLV(A) is uniquely determined byf|V :V →V ⊗A, natural transformations are therefore in bijection with ρ:V →V ⊗Athat induces A-linear automorphism onV ⊗A.
But we do not just want natural transformations; we also require that each Φ(R) : G(R)→GLV(R) is a group homomorphism, i.e.
Φ(R)(gh) = Φ(R)(g)◦Φ(R)(h), for all g, h∈ G(R). (7) Lettingρ be the restriction ofρe:= Φ(A)(idA) to V, we get that
Φ(R)(gh) = Φ(R)((g, h)◦∆) = GLV((g, h)◦∆)(ρe) = (GLV((g, h))◦GLV(∆))(ρe),
with ∆ the comultiplication onA. But (GLV((g, h))◦GLV(∆))(ρe) is simply the automor-
phism induced by (id⊗(g, h))◦(id⊗∆)◦ρ:V →V ⊗R. Similarly, Φ(R)(g) and Φ(R)(h) are induced by (id⊗g)◦ρ and (id⊗h)◦ρ respectively. To determine their restriction, take an arbitraryv∈V. We first see that Φ(R)(h) restricted to V sends
v7−→ρ Xvi⊗ai
id⊗h
7−→ Xvi⊗h(ai) =
X
h(ai)(vi⊗1). Then Φ(R)(g) restricted to V sends
X vi ρ 7−→X i X j vij⊗bj id⊗g 7−→X i X j vij⊗g(bj).
Thus the restriction of Φ(R)(g)◦Φ(R)(h) to V maps
v7→X i h(ai) X j vij⊗g(bj) = X i X j vij⊗(g(bj)h(ai)) = (id⊗(g, h)) X i X j vij⊗bj⊗ai .
But we also have that
((ρ⊗id)◦ρ)(v) = (ρ⊗id)(Xvi⊗ai) =X i (X j vij ⊗bj)⊗ai =X i (X j vij ⊗bj ⊗ai).
Thus, (id⊗(g, h))◦(ρ⊗id)◦ρis the restriction of Φ(R)(g)◦Φ(R)(h) toV. By the uniqueness of the restriction, equation (7) holds if and only if the following equation holds:
(id⊗(g, h))◦(id⊗∆)◦ρ= (id⊗(g, h))◦(ρ⊗id)◦ρ.
But we need this to be true for allR, in particular withR=A andg=h= idA. So this is equivalent to
(id⊗∆)◦ρ= (ρ⊗id)◦ρ. (8) Note ρ = Φ(A)(idA) that satisfy this equation would have its corresponding Φ satisfying equation (7), which implies that images of Φ(R) are automorphisms. In particularρbeing an image of Φ(A) will be an automorphism. Thus equation (8) gives us an equivalent condition onρ:V →V ⊗A so that the corresponding Φ :G →GLV is a group homomorphism:
5.1. LINEAR REPRESENTATIONS AND COMODULES 55
Theorem 5.1.2. For G = Speck(A) an affine group scheme, its linear representations
Φ :G →GLV over V are in bijection with k-linear mapsρ:V →V ⊗A satisfying
V V ⊗A
V ⊗A V ⊗A⊗A.
ρ
ρ ρ⊗id
id⊗∆
We say that a k-module V equipped with k-linear map ρ : V → V ⊗A satisfying the above commutative diagram is acomoduleover a Hopfk-algebraA. One immediate example of aA-comodule is withV =Aandρ= ∆, where the required diagram is obtained from the coassociativity ofA. The corresponding representation is called theregular representation
of G ∼= Spec(A). Asubcomodule W of V is then a k-submodule ofV withρ(W)⊆W ⊗A. In terms of representations, this then corresponds to a subrepresentation ofV.
Remark 5.1.3. In most texts, the definition of a comodule would include a second condition:
V V ⊗A V V ⊗k. ρ = id⊗ ∼ = (9)
But note that this condition is implied. To see this, the commutative diagram
V V ⊗A
V ⊗A V ⊗A⊗A,
ρ
ρ ρ⊗id
id⊗∆
given by our definition holds if and only if the induced map Φ :G → GLV is a morphism of group-functor. But we know that a group functor morphism preserves identity, which implies that
Φ(R)(u◦) = idV⊗R
for all k-algebrasR, where u:k→R is the ring map providing the algebra structure. But this says that
V V ⊗A V ⊗k V ⊗R ρ ∼ = id⊗ id⊗(u◦) id⊗u
commutes for all R, which holds if and only if diagram (9) commutes.
When V is free of finite rank n with basis {ei}ni=1, we can identify GLV with GLn ∼=
Spec(k[xij,det−1]) =: Spec(M). Thus the representation Φ : G ∼= Spec(A) → GLn cor- responds to a Hopf-algebra homomorphism Φ# : M → A. With respect to this basis,
ρ:V → V ⊗A is determined uniquely by the images of the basis elements ei. If we write
ρ(ej) = Pei⊗aij, then Φ(A)(idA) = Φ# : M → A would be represented by the matrix (aij), with xij 7→ aij. Since we have that ∆M(xij) = Pkxik ⊗xkj by preservation of comultiplication we get the following identity:
∆A(aij) = ∆A(Φ#(xij)) = Φ#(∆M(xij)) =
X
k
aik⊗akj. (10)