CHAPTER 4 The Infinitesimal Theory

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4. Derivations and Lie Algebras of Affine Algebraic Groups

Lemma and Definition 7.4.1. Let G : 𝕂-cAlg ⟶ Set be a group valued functor. The kernel Lie (G)(R) of the sequence

[figure: 0; Lie (G)(R); G(R(δ)); G(p); G(j); G(R); 0]

is called the Lie algebra of G and is a group valued functor in R.

Proof. For every algebra homomorphism f : R ⟶ S the following diagram of groups commutes

[figure: 0; Lie (G)(R); G(R(δ)); G(p); G(j); G(R); 0; G(f(δ)); G(f); 0; Lie (G)(S); G(S(δ)); G(p); G(j); G(S); 0]

Proposition 4.4.2. Let G : 𝕂-cAlg ⟶ Set be a group valued functor with multiplication ∗. Then there are functorial operations

G(R) × Lie (G)(R) ∋ (g, x) ↦ g · x ∈ Lie (G)(R)
R × Lie (G)(R) ∋ (a, x) ↦ ax ∈ Lie (G)(R)

such that

g · (x + y) = g · x + g · y,
h · (g · x) = (h ∗ g) · x,
a(x + y) = ax + ay,
(ab)x = a(bx),
g · (ax) = a(g · x).

Proof. First observe that the composition + on Lie (G)(R) is induced by the multiplication ∗ of G(R(δ)) so it is not necessarily commutative.

We define g·x := G(j)(g)∗x∗G(j)(g)^{−1}. Then G(p)(g·x) = G(p)G(j)(g)∗G(p)(x)∗ G(p)G(j)(g)^{−1} = g ∗ 1 ∗ g^{−1} = 1 hence g · x ∈ Lie (G)(R).

Now let a ∈ R. To define a : Lie (G)(R) ⟶ Lie (G)(R) we use u_{a} : R(δ) ⟶ R(δ) defined by u_{a}(δ) := aδ and thus u_{a}(b+cδ) := b+acδ. Obviously u_{a} is a homomorphism of R-algebras. Furthermore we have pu_{a} = p and u_{a}j = j. Thus we get a commutative diagram

[figure: 0; Lie (G)(R); G(R(δ)); G(p); G(j); G(R); 0; a; G(u_{a}); id; 0; Lie (G)(R); G(R(δ)); G(p); G(j); G(R); 0]

that defines a group homomorphism a : Lie (G)(R) ⟶ Lie (G)(R) on the kernel of the exact sequences. In particular we have then a(x + y) = ax + ay.

Furthermore we have u_{ab} = u_{a}u_{b} hence (ab)x = a(bx).

The next formula follows from g · (x + y) = G(j)(g) ∗ x ∗ y ∗ G(j)(g)^{−1} = G(j)(g) ∗ x ∗ G(j)(g)^{−1} ∗ G(j)(g) ∗ y ∗ G(j)(g)^{−1} = g · x + g · y.

We also see (h∗g)·x = G(j)(h∗g)∗x∗G(j)(h∗g)^{−1} = G(j)(h)∗G(j)(g)∗x∗G(j)(g)^{−1}∗ G(j)(h)^{−1} = h · (g · x). Finally we have g · (ax) = G(j)(g) ∗ G(u_{a})(x) ∗ G(j)(g^{−1}) = G(u_{a})(G(j)(g) ∗ x ∗ G(j)(g^{−1})) = a(g · x).

Proposition 4.4.3. Let G = 𝕂-cAlg(H, -) be an affine algebraic group. Then Lie (G)(𝕂) ≅ Lie(H^{o}) as additive groups. The isomorphism is compatible with the operations given in 4.4.2 and 4.3.6.

Proof. We consider the following diagram

[figure: 0; Lie (G)(𝕂); 𝕂-cAlg(H, 𝕂(δ)); p; 𝕂-cAlg(H, 𝕂); 0; e; ω; ≅; 0; Lie(H^{o}); G_{𝕂(δ)}(H^{o} ⊗ 𝕂(δ)); e^{δ-}; p; G(H^{o}); 0]

We know by definition that the top sequence is exact. The bottom sequence is exact by Corollary 4.3.9.

Let f ∈ 𝕂-cAlg(H, 𝕂). Since Ker(f) is an ideal of codimension 1 we get f ∈ H^{o}. The map f is an algebra homomorphism iff ⟨f, ab⟩ = ⟨f ⊗ f, a ⊗ b⟩ and ⟨f, 1⟩ = 1 iff Δ_{H^{o}}(f) = f ⊗ f and ε_{H^{o}}(f) = 1 iff f ∈ G(H^{o}). Hence we get the right hand vertical isomorphism 𝕂-cAlg(H, 𝕂) ≅ G(H^{o}).

Consider an element f ∈ 𝕂-cAlg(H, 𝕂(δ)) ⊆ Hom(H, 𝕂(δ)). It can be written as f = f_{0}+f_{1}δ with f_{0}, f_{1} ∈ Hom(H, 𝕂). The linear map f is an algebra homomorphism iff f_{0} : H ⟶ 𝕂 is an algebra homomorphism and f_{1} satisfies f_{1}(1) = 0 and f_{1}(ab) = f_{0}(a)f_{1}(b) + f_{1}(a)f_{0}(b). In fact we have f(1) = f_{0}(1) + f_{1}(1)δ = 1 iff f_{0}(1) = 1 and f_{1}(1) = 0 (by comparing coefficients). Furthermore we have f(ab) = f(a)f(b) iff f_{0}(ab) + f_{1}(ab)δ = (f_{0}(a) + f_{1}(a)δ)(f_{0}(b) + f_{1}(b)δ) = f_{0}(a)f_{0}(b) + f_{0}(a)f_{1}(b)δ + f_{1}(a)f_{0}(b)δ iff f_{0}(ab) = f_{0}(a)f_{0}(b) and f_{1}(ab) = f_{0}(a)f_{1}(b) + f_{1}(a)f_{0}(b).

Since f_{0} is an algebra homomorphism we have as above f_{0} ∈ H^{o}. For f_{1} we have (b ⇀ f_{1})(a) = f_{1}(ab) = f_{0}(a)f_{1}(b) + f_{1}(a)f_{0}(b) = (f_{1}(b)f_{0} + f_{0}(b)f_{1})(a) hence (b ⇀ f_{1}) = f_{1}(b)f_{0} +f_{0}(b)f_{1} ∈ 𝕂f_{0} +𝕂f_{1}, a two dimensional subspace. Thus f_{1} ∈ H^{o}.

In the following computations we will identify (H^{o} ⊗ 𝕂(δ)) ⊗_{𝕂(δ)}(H^{o} ⊗ 𝕂(δ)) with H^{o} ⊗ H^{o} ⊗ |K(δ).

Let f = f_{0} + f_{1}δ = f_{0} ⊗ 1 + f_{1} ⊗ δ ∈ H^{o} ⊕ H^{o}δ = H^{o} ⊗ 𝕂(δ). Then f is a homomorphism of algebras iff f(ab) = f(a)f(b) and f(1) = 1 iff f_{0}(ab) = f_{0}(a)f_{0}(b) and f_{1}(ab) = f_{0}(a)f_{1}(b)+f_{1}(a)f_{0}(b) and f_{0}(1) = 1 and f_{1}(1) = 0 iff Δ_{H^{o}}(f_{0}) = f_{0} ⊗f_{0} and Δ_{H^{o}}(f_{1}) = f_{0}⊗f_{1}+f_{1}⊗f_{0} and ε_{H^{o}}(f_{0}) = 1 and ε_{H^{o}}(f_{1}) = 0 iff (Δ_{H^{o}} ⊗id_{𝕂(δ)})(f_{0}⊗ 1+f_{1} ⊗δ) = f_{0} ⊗f_{0} ⊗1+f_{0} ⊗f_{1} ⊗δ+f_{1} ⊗f_{0} ⊗δ = (f_{0} ⊗1+f_{1} ⊗δ)⊗_{𝕂(δ)}(f_{0} ⊗1+f_{1} ⊗δ) and (ε_{H^{o}} ⊗ id_{𝕂(δ)})(f_{0} ⊗ 1 + f_{1} ⊗ δ) = 1 ⊗ 1 iff (Δ_{H^{o}} ⊗ id_{𝕂(δ)})(f) = f ⊗_{𝕂(δ)} f and (ε_{H^{o}} ⊗ id_{𝕂(δ)})(f) = 1 iff f ∈ G_{𝕂(δ)}(H^{o} ⊗ 𝕂(δ)).

Hence we have a bijective map ω : 𝕂-cAlg(H, 𝕂(δ)) ∋ f = f_{0} + f_{1}δ ↦ f_{0} ⊗ 1 + f_{1} ⊗ δ ∈ G_{𝕂(δ)}(H^{o} ⊗ 𝕂(δ)). Since the group multiplication in 𝕂-cAlg(H, 𝕂(δ)) ⊆ Hom(H, 𝕂(δ)) is the convolution ∗ and the group multiplication in G_{𝕂(δ)}(H^{o}⊗𝕂(δ)) ⊆ H^{o} ⊗ 𝕂(δ) is the ordinary algebra multiplication, where the multiplication of H^{o} again is the convolution, it is clear that ω is a group homomorphism. Furthermore the right hand square of the above diagram commutes. Thus we get an isomorphism e : Lie(H^{o}) ⟶ Lie (G)(𝕂) on the kernels. This map is defined by e(d) = 1 + dδ ∈ 𝕂-cAlg(H, 𝕂(δ)).

To show that this isomorphism is compatible with the actions of 𝕂 resp. G(H^{o}) let α ∈ 𝕂, a ∈ H, and d ∈ Lie(H^{o}). We have e(αd)(a) = ε(a) + αd(a)δ = u_{α}(ε(a) + d(a)δ) = (u_{α} ∘ (1 + dδ))(a) = (u_{α} ∘ e(d))(a) = (αe(d))(a) hence e(αd) = αe(d).

Furthermore let g ∈ G(H^{o}) = 𝕂-cAlg(H, 𝕂), a ∈ H, and d ∈ Lie(H^{o}). Then we have e(g · d)(a) = e(gdg^{−1})(a) = (1 + gdg^{−1}δ)(a) = ε(a) + gdg^{−1}(a)δ = ∑ g(a_{(1)})ε(a_{(2)})gS(a_{(3)}) + ∑ g(a_{(1)})d(a_{(2)})gS(a_{(3)})δ = ∑ g(a_{(1)})e(d)(a_{(2)})gS(a_{(3)}) = (j ∘ g ∗ e(d) ∗ j ∘ g^{−1})(a) = (g · e(d))(a) hence e(g · d) = g · e(d).

Proposition 4.4.4. Let H be a Hopf algebra and let I := Ker(ε). Then Der_{ε}(H, -) : Vec ⟶ Vec is representable by I/I^{2} and d : H ⟶^{1−ε} I ⟶^{ν} I/I^{2}, in particular

Der_{ε}(H, -) ≅ Hom(I/I^{2}, -)  and  Lie(H^{o}) ≅ Hom(I/I^{2}, 𝕂).

Proof. Because of ε(id −uε)(a) = ε(a) − εuε(a) = 0 we have Im(id −ε) ⊆ I. Let i ∈ I. Then we have i = i−ε(i) = (id −ε)(i) hence Im(id −ε) = Ker(ε). We have I^{2} ∋ (id −ε)(a)(id −ε)(b) = ab − ε(a)b − aε(b) + ε(a)ε(b) = (id −ε)(ab) − ε(a)(id −ε)(b) − (id −ε)(b). Hence we have in I/I^{2} the equation (id −ε)(ab) = ε(a)(id −ε)(b) + (id −ε)(a)ε(b) so that ν(id −ε) : H ⟶ I ⟶ I/I^{2} is an ε-derivation.

Now let D : H ⟶ M be an ε-derivation. Then D(1) = D(1 1) = 1D(1) + D(1)1 hence D(1) = 0. It follows D(a) = D(id −ε)(a). From ε(I) = 0 we get D(I^{2}) ⊆ ε(I)D(I) + D(I)ε(I) = 0 hence there is a unique factorization

[figure: H; I; id −ε; I/I^{2}; ν; D; D; M.; f]

Corollary 4.4.5. Let H be a Hopf algebra that is finitely generated a s an algebra. Then Lie(H^{o}) is finite dimensional.

Proof. Let H = 𝕂⟨a_{1}, …, a_{n}⟩. Since H = 𝕂 ⊕ I we can choose a_{1} = 1 and a_{2}, …, a_{n} ∈ I. Thus any element in i ∈ I can be written as ∑ α_{J}a_{j_{1}} … a_{j_{k}} so that I/I^{2} = 𝕂\overline{a_{2}} + … + \overline{a_{n}}. This gives the result.

Proposition 4.4.6. Let H be a commutative Hopf algebra and _{H}M be an Hmodule. Then we have Ω_{H} ≅ H ⊗ I/I^{2} and d : H ⟶ H ⊗ I/I^{2} is given by d(a) = ∑ a_{(1)} ⊗ \overline{(id −ε)(a_{(2)})}.

Proof. Consider the algebra B := H ⊕ M with (a, m)(a′, m′) = (aa′, am′ + a′m). Let G = 𝕂-cAlg(H, -). Then we have G(B) ⊆ Hom(H, B) ≅ Hom(H, H) ⊕ Hom(H, M). An element (φ, D) ∈ Hom(H, B) is in G(B) iff (φ, D)(1) = (φ(1), D(1)) = (1, 0), hence φ(1) = 1 and D(1) = 0, and (φ(ab), D(ab)) = (φ, D)(ab) = (φ, D)(a)(φ, D)(b) = (φ(a), D(a))(φ(b), D(b)) = (φ(a)φ(b), φ(a)D(b) + D(a)φ(b), hence φ(ab) = φ(a)φ(b) and D(ab) = φ(a)D(b) + D(a)φ(b). So (φ, D) is in G(B) iff φ ∈ G(H) and D is a φ-derivation. The ∗-multiplication in Hom(H, B) is given by (φ, D) ∗ (φ′, D′) = (φ ∗ φ′, φ ∗ D′ + D ∗ φ′) by applying this to an element a ∈ H. Since (φ, 0) ∈ G(B) and (uε, D) ∈ G(B) for every ε-derivation D, there is a bijection Der_{ε}(H, M) ≅ {(uε, D_{ε}) ∈ G_{ε}(B)} ≅ {(1_{H}, D_{1}) ∈ G_{1}(B)} ≅ Der_{𝕂}(H, M) by (uε, D_{ε}) ↦ (1, 0) ∗ (uε, D_{ε}) = (1, 1 ∗ D_{ε}) ∈ G_{1}(B) with inverse map (1, D_{1}) ↦ (S, 0) ∗ (1, D_{1}) = (uε, S ∗ D_{1}) ∈ G_{ε}(B). Hence we have isomorphisms Der_{𝕂}(H, M) ≅ Der_{ε}(H, M) ≅ Hom(I/I^{2}, M) ≅ Hom_{H}(H ⊗ I/I^{2}, M).

The universal ε-derivation for vector spaces is \overline{id −ε} : A ⟶ I/I^{2}. The universal ε-derivation for H-modules is D_{ε}(a) = 1 ⊗ \overline{(id −ε)(a)} ∈ A ⊗ I/I^{2}. The universal 1-derivation for H-modules is 1 ∗ D_{ε} with (1 ∗ D_{ε})(a) = ∑ a_{(1)} ⊗ \overline{(id −ε)(a_{(2)})} ∈ A ⊗ I/I^{2}.