CHAPTER 4 The Infinitesimal Theory

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3. The Lie Algebra of Primitive Elements

Lemma 4.3.1. Let H be a Hopf algebra and H^{o} be its Sweedler dual. If d ∈ Der_{𝕂}(H,_{ε}𝕂_{ε}) ⊆ Hom(H, 𝕂) is a derivation then d is a primitive element of H^{o}. Furthermore every primitive element d ∈ H^{o} is a derivation in Der_{𝕂}(H,_{ε}𝕂_{ε}).

Proof. Let d : H ⟶ 𝕂 be a derivation and let a, b ∈ H. Then (b ⇀ d)(a) = d(ab) = ε(a)d(b) + d(a)ε(b) = (d(b)ε + ε(b)d)(a) hence (b ⇀ d) = d(b)ε + ε(b)d. Consequently we have Hd = (H ⇀ d) ⊆ 𝕂ε + 𝕂d so that dim Hd ≤ 2 < ∞. This shows d ∈ H^{o}. Furthermore we have ⟨Δd, a ⊗ b⟩ = ⟨d, ab⟩ = d(ab) = d(a)ε(b) + ε(a)d(b) = ⟨d ⊗ ε, a ⊗ b⟩ + ⟨ε ⊗ d, a ⊗ b⟩ = ⟨1_{H^{o}} ⊗ d + d ⊗ 1_{H^{o}}, a ⊗ b⟩ hence Δ(d) = d ⊗ 1_{H^{o}} + 1_{H^{o}} ⊗ d so that d is a primitive element in H^{o}.

Conversely let d ∈ H^{o} be primitive. then d(ab) = ⟨Δ(d), a ⊗ b⟩ = d(a)ε(b) + ε(a)d(b).

Proposition and Definition 4.3.2. Let H be a Hopf algebra. The set of primitive elements of H will be denoted by Lie(H) and is a Lie algebra. If char(𝕂) = p > 0 then Lie(H) is a restricted Lie algebra or a p-Lie algebra.

Proof. Let a, b ∈ H be primitive elements. Then Δ([a, b]) = Δ(ab − ba) = (a⊗1+1⊗a)(b⊗1+1⊗b)−(b⊗1+1⊗b)(a⊗1+1⊗a) = (ab−ba)⊗1+1⊗(ab−ba) hence Lie(H) ⊆ H^{L} is a Lie algebra. If the characteristic of 𝕂 is p > 0 then we have (a ⊗ 1 + 1 ⊗ a)^{p} = a^{p} ⊗ 1 + 1 ⊗ a^{p}. Thus Lie(H) is a restricted Lie subalgebra of H^{L} with the structure maps [a, b] = ab − ba and a^{[p]} = a^{p}.

Corollary 4.3.3. Let H be a Hopf algebra. Then the set of left translation invariant derivations D : H ⟶ H is a Lie algebra under [D, D′] = DD′ − D′D. If char = p then these derivations are a restricted Lie algebra with D^{[p]} = D^{p}.

Proof. The map Ψ : H^{o} ⟶ H^{∗} ⟶^{Φ} End(H) is a homomorphism of algebras by 4.2.6. Hence Ψ(d ∗ d′ − d′ ∗ d) = Φ(d ∗ d′ − d′ ∗ d) = Φ(d)Φ(d′) − Φ(d′)Φ(d). If d is a primitive element in H^{o} then by 4.2.7 and 4.3.1 the image D := Ψ(d) in End(H) is a left translation invariant derivation and all left translation invariant derivations are of this form. Since [d, d′] = d ∗ d′ − d′ ∗ d is again primitive we get that [D, D′] = DD′ − D′D is a left translation invariant derivation so that the set of left translation invariant derivations Der_{𝕂}^{H}(H, H) is a Lie algebra resp. a restricted Lie algebra.

Definition 4.3.4. Let H be a Hopf algebra. An element c ∈ H is called cocommutative if τΔ(c) = Δ(c), i. e. if ∑ c_{(1)} ⊗ c_{(2)} = ∑ c_{(2)} ⊗ c_{(1)}. Let C(H) := {c ∈ H|c is cocommutative }.

Let G(H) denote the set of group like elements of H.

Lemma 4.3.5. Let H be a Hopf algebra. Then the set of cocommutative elements C(H) is a subalgebra of H and the group like elements G(H) form a linearly independent subset of C(H). Furthermore G(H) is a multiplicative subgroup of the group of units U(C(H)).

Proof. It is clear that C(H) is a linear subspace of H. If a, b ∈ C(H) then Δ(ab) = Δ(a)Δ(b) = (τΔ)(a)(τΔ)(b) = τ(Δ(a)Δ(b)) = τΔ(ab) and Δ(1) = 1 ⊗ 1 = τΔ(1). Thus C(H) is a subalgebra of H.

The group like elements obviously are cocommutative and form a multiplicative group, hence a subgroup of U(C(H)). They are linearly independent by Lemma 2.1.14.

Proposition 4.3.6. Let H be a Hopf algebra with S^{2} = id_{H}. Then there is a left module structure

C(H) ⊗ Lie(H) ∋ c ⊗ a ↦ c · a ∈ Lie(H)

with c · a := ∇_{H}(∇_{H} ⊗ 1)(1 ⊗ τ)(1 ⊗ S ⊗ 1)(Δ ⊗ 1)(c ⊗ a) = ∑ c_{(1)}aS(c_{(2)}) such that

c · [a, b] = ∑[c_{(1)} · a, c_{(2)} · b].

In particular G(H) acts by Lie automorphisms on Lie(H).

Proof. The given action is actually the action H ⊗ H ⟶ H with h · a = ∑ h_{(1)}aS(h_{(2)}), the so-called adjoint action.

We first show that the given map has image in Lie(H). For c ∈ C(H) and a ∈ Lie(H) we have Δ(c · a) = Δ(∑ c_{(1)}aS(c_{(2)})) = ∑ Δ(c_{(1)})(a ⊗ 1 + 1 ⊗ a)Δ(S(c_{(2)})) = ∑ Δ(c_{(1)})(a⊗1)Δ(S(c_{(2)}))+∑ Δ(c_{(2)})(1⊗a)Δ(S(c_{(1)})) = ∑ c_{(1)}aS(c_{(4)})⊗c_{(2)}S(c_{(3)})+ ∑ c_{(3)}S(c_{(2)}) ⊗ c_{(4)}aS(c_{(1)}) = c · a ⊗ 1 + 1 ⊗ c · a since c is cocommutative, S^{2} = id_{H} and a is primitive.

We show now that Lie(H) is a C(H)-module. (cd) · a = ∑ c_{(1)}d_{(1)}aS(c_{(2)}d_{(2)}) = ∑ c_{(1)}d_{(1)}aS(d_{(2)})S(c_{(2)}) = c · (d · a). Furthermore we have 1 · a = 1aS(1) = a.

To show the given formula let a, b ∈ Lie(H) and c ∈ C(H). Then c · [a, b] = ∑ c_{(1)}(ab−ba)S(c_{(2)}) = ∑ c_{(1)}aS(c_{(2)})c_{(3)}bS(c_{(4)})−∑ c_{(1)}bS(c_{(2)})c_{(3)}aS(c_{(4)}) = ∑(c_{(1)}· a)(c_{(2)}·b)−∑(c_{(1)}·b)(c_{(2)}·a) = ∑[c_{(1)}·a, c_{(2)}·b] again since c ∈ C(H) is cocommutative.

Now let g ∈ G(H). Then g · a = gaS(g) = gag^{−1} since S(g) = g^{−1} for any group like element. Furthermore g · [a, b] = [g · a, g · b] hence g defines a Lie algebra automorphism of Lie(H).

Problem 4.3.1. Show that the adjoint action H ⊗H ∋ h⊗a ↦ ∑ h_{(1)}aS(h_{(2)}) ∈ H makes H an H-module algebra.

Definition and Remark 4.3.7. The algebra 𝕂(δ) = 𝕂[δ]/(δ^{2}) is called the algebra of dual numbers. Observe that 𝕂(δ) = 𝕂 ⊕ 𝕂δ as a 𝕂-module.

We consider δ as a ”small quantity“ whose square vanishes.

The maps p : 𝕂(δ) ⟶ K with p(δ) = 0 and j : 𝕂 ⟶ 𝕂(δ) are algebra homomorphism satisfying pj = id.

Let 𝕂(δ, δ′) := 𝕂[δ, δ′]/(δ^{2}, δ′^{2}). Then 𝕂(δ, δ′) = 𝕂 ⊕ 𝕂δ ⊕ 𝕂δ′ ⊕ 𝕂δδ′. The map 𝕂(δ) ∋ δ ↦ δδ′ ∈ 𝕂(δ, δ′) is an injective algebra homomorphism. Furthermore for every α ∈ 𝕂 we have an algebra homomorphism φ_{α} : 𝕂(δ) ∋ δ ↦ αδ ∈ 𝕂(δ).

These algebra homomorphisms induce algebra homomorphisms H ⊗ 𝕂(δ) ⟶ H ⊗ 𝕂(δ) resp. H ⊗ 𝕂(δ) ⟶ H ⊗ 𝕂(δ, δ′) for every Hopf algebra H.

Proposition 4.3.8. The map

e^{δ-} : Lie(H) ⟶ H ⊗ 𝕂(δ) ⊆ H ⊗ 𝕂(δ, δ′)

with e^{δa} := 1 + a ⊗ δ = 1 + δa is called the exponential map and satisfies

e^{δ(a+b)} = e^{δa}e^{δb},
e^{δαa} = φ_{α}(e^{δa}),
e^{δδ′[a,b]} = e^{δa}e^{δ′b}(e^{δa})^{−1}(e^{δ′b})^{−1}.

Furthermore all elements e^{δa} ∈ H ⊗ 𝕂(δ) are group like in the 𝕂(δ)-Hopf algebra H ⊗ 𝕂(δ).

Proof. 1. e^{δ(a+b)} = (1 + δ(a + b)) = (1 + δa)(1 + δb) = e^{δa}e^{δb}.

2. e^{δαa} = 1 + δαa = φ_{α}(1 + δa) = φ_{α}(e^{δa}).

3. Since (1+δa)(1−δa) = 1 we have (e^{δa}) = 1−δa. So we get e^{δδ′[a,b]} = 1+δ[a, b] = 1 + δ(a − a) + δ′(b − b) + δδ′(ab − ab − ba + ab) = (1 + δa)(1 + δ′b)(1 − δa)(1 − δ′b) = e^{δa}e^{δ′b}(e^{δa})^{−1}(e^{δ′b})^{−1}.

4. Δ_{𝕂(δ)}(e^{δa}) = Δ(1 + a ⊗ δ) = 1 ⊗_{𝕂(δ)} 1 + (a ⊗ 1 + 1 ⊗ a) ⊗ δ = 1 ⊗_{𝕂(δ)} 1 + δa ⊗_{𝕂(δ)} 1 + 1 ⊗_{𝕂(δ)} δa + δa ⊗_{𝕂(δ)} δa = (1 + δa) ⊗_{𝕂(δ)} (1 + δa) = e^{δa} ⊗_{𝕂(δ)} e^{δa} and ε_{𝕂(δ)}(e^{δa}) = ε_{𝕂(δ)}(1 + δa) = 1 + δε(a) = 1.

Corollary 4.3.9. (Lie(H), e^{δ-}) is the kernel of the group homomorphism p : G_{𝕂(δ)}(H ⊗ 𝕂(δ)) ⟶ G(H).

Proof. p = 1 ⊗ p : H ⊗ 𝕂(δ) ⟶ H ⊗ 𝕂 = H is a homomorphism of 𝕂-algebras. We show that it preserves group like elements. Observe that group like elements in H ⊗ 𝕂(δ) are defined by the Hopf algebra structure over 𝕂(δ). Let g ∈ G_{𝕂(δ)}(H ⊗ 𝕂(δ)). Then (Δ_{H} ⊗ 1)(g) = g ⊗_{𝕂(δ)} g and (ε_{H} ⊗ 1)(g) = 1 ∈ 𝕂(δ).

Since p : 𝕂(δ) ⟶ 𝕂 is an algebra homomorphism the following diagram commutes

[figure: (H ⊗ 𝕂) ⊗ (H ⊗ 𝕂); H ⊗ H ⊗ 𝕂.; ≅; (H ⊗ 𝕂(δ)) ⊗_{𝕂(δ)} (H ⊗ 𝕂(δ)); H ⊗ H ⊗ 𝕂(δ); ≅; (1⊗p)⊗(1⊗p); 1⊗p]

We identify elements along the isomorphisms. Thus we get (Δ_{H} ⊗ 1_{𝕂})(1_{H} ⊗ p)(g) = (1_{H⊗H} ⊗ p)(Δ_{H} ⊗ 1_{𝕂(δ)})(g) = ((1_{H} ⊗ p) ⊗_{𝕂(δ)} (1_{H} ⊗ p))(g ⊗_{𝕂(δ)} g) = (1_{H} ⊗ p)(g) ⊗ (1_{H} ⊗ p)(g), so that 1_{H} ⊗ p : G_{𝕂(δ)}(H ⊗ 𝕂(δ)) ⟶ G(H). Now we have (1_{H} ⊗ p)(gg′) = (1_{H} ⊗ p)(g)(1_{H} ⊗ p)(g′) so that 1_{H} ⊗ p is a group homomorphism.

Now let g = g_{0} ⊗ 1 + g_{1} ⊗ δ ∈ G_{𝕂(δ)}(H ⊗ 𝕂(δ)) ⊆ H ⊗ 𝕂 ⊕ H ⊗ 𝕂δ. Then we have (1_{H} ⊗ p)(g) = 1 iff g_{0} = 1 iff g = 1_{H} ⊗ 1_{𝕂(δ)} + g_{1} ⊗ δ. Furthermore we have

Δ_{H⊗𝕂(δ)}(g) = g ⊗_{𝕂(δ)} g ⟺

1_{H} ⊗ 1_{H} ⊗ 1_{𝕂(δ)} + Δ_{H}(g_{1}) ⊗ δ = (1_{H} ⊗ 1_{𝕂(δ)} + g_{1} ⊗ δ) ⊗_{𝕂(δ)} (1_{H} ⊗ 1_{𝕂(δ)} + g_{1} ⊗ δ)

= 1_{H} ⊗ 1_{H} ⊗ 1_{𝕂(δ)} + (g_{1} ⊗ 1_{H} + 1_{H} ⊗ g_{1}) ⊗ δ ⟺

Δ_{H}(g_{1}) = g_{1} ⊗ 1_{H} + 1_{H} ⊗ g_{1}.

Similarly we have ε_{𝕂(δ)})(g) = 1 iff 1 ⊗ 1 + ε(g_{1}) ⊗ δ = 1 iff ε(g_{1}) = 0.