CHAPTER 2 Hopf Algebras, Algebraic, Formal, and Quantum Groups

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7. Duality of Hopf Algebras

In 2.4.8 we have seen that the dual Hopf algebra H^{∗} of a finite dimensional Hopf algebra H satisfies certain relations w.r.t. the evaluation map. The multiplication of H^{∗} is derived from the comultiplication of H and the comultiplication of H^{∗} is derived from the multiplication of H.

This kind of duality is restricted to the finite-dimensional situation. Nevertheless one wants to have a process that is close to the finite-dimensional situation. This short section is devoted to several approaches of duality for Hopf algebras.

First we use the relations of the finite-dimensional situation to give a general definition.

Definition 2.7.1. Let H and L be Hopf algebras. Let

ev : L ⊗ H ∋ a ⊗ h ↦ ⟨a, h⟩ ∈ 𝕂

be a bilinear form satisfying

⟨a ⊗ b, ∑ h_{(1)} ⊗ h_{(2)}⟩ = ⟨ab, h⟩, ⟨1, h⟩ = ε(h) (1)

⟨∑ a_{(1)} ⊗ a_{(2)}, h ⊗ j⟩ = ⟨a, hj⟩, ⟨a, 1⟩ = ε(a) (2)

⟨a, S(h)⟩ = ⟨S(a), h⟩ (3)

Such a map is called a weak duality of Hopf algebras. The bilinear form is called left (right) nondegenerate if ⟨a, H⟩ = 0 implies a = 0 (⟨L, h⟩ = 0 implies h = 0). A duality of Hopf algebras is a weak duality that is left and right nondegenerate.

Remark 2.7.2. If H is a finite dimensional Hopf algebra then the usual evaluation ev : H^{∗} ⊗ H ⟶ 𝕂 defines a duality of Hopf algebras.

Remark 2.7.3. Assume that ev : L ⊗ H ⟶ 𝕂 defines a weak duality. By A.4.15 we have isomorphisms Hom(L ⊗ H, 𝕂) ≅ Hom(L, Hom(H, 𝕂)) and Hom(L ⊗ H, 𝕂) ≅ Hom(H, Hom(L, 𝕂)). Denote the homomorphisms associated with ev : L ⊗ K ⟶ 𝕂 by φ : L ⟶ Hom(H, 𝕂) resp. ψ : H ⟶ Hom(L, 𝕂). They satisfy φ(a)(h) = ev(a ⊗ h) = ψ(h)(a).

ev : L ⊗ K ⟶ 𝕂 is left nondegenerate iff φ : L ⟶ Hom(H, 𝕂) is injective. ev : L ⊗ K ⟶ 𝕂 is right nondegenerate iff ψ : H ⟶ Hom(L, 𝕂) is injective.

Lemma 2.7.4. 1. The bilinear form ev : L ⊗ H ⟶ 𝕂 satisfies (1) if and only if φ : L ⟶ Hom(H, 𝕂) is a homomorphism of algebras.

2. The bilinear form ev : L ⊗ H ⟶ 𝕂 satisfies (2) if and only if ψ : H ⟶ Hom(L, 𝕂) is a homomorphism of algebras.

Proof. ev : L⊗H ⟶ 𝕂 satisfies the right equation of (1) iff φ(ab)(h) = ⟨ab, h⟩ = ⟨a ⊗ b, ∑ h_{(1)} ⊗ h_{(2)}⟩ = ∑⟨a, h_{(1)}⟩⟨b, h_{(2)}⟩ = ∑ φ(a)(h_{(1)})φ(b)(h_{(2)}) = (φ(a) ∗ φ(b))(h) by the definition of the algebra structure on Hom(H, 𝕂). ev : L ⊗ H ⟶ 𝕂 satisfies the left equation of (1) iff φ(1)(h) = ⟨1, h⟩ = ε(h). The second part of the Lemma follows by symmetry.

Example 2.7.5. There is a weak duality between the quantum groups 𝕊𝕃_{q}(2) and U_{q}(sl(2)). (Kassel: Chapter VII.4).

Proposition 2.7.6. Let ev : L ⊗ H ⟶ 𝕂 be a weak duality of Hopf algebras. Let I := Ker(φ : L ⟶ Hom(H, 𝕂)) and J := Ker(ψ : H ⟶ Hom(L, 𝕂)). Let \overline{L} := L/I and \overline{H} := H/J. Then \overline{L} and \overline{H} are Hopf algebras and the induced bilinear form \overline{ev} : \overline{L} ⊗ \overline{H} ⟶ 𝕂 is a duality.

Proof. First observe that I and J are two sided ideals hence \overline{L} and \overline{H} are algebras. Then ev : L ⊗ H ⟶ 𝕂 can be factored through \overline{ev} : \overline{L} ⊗ \overline{H} ⟶ 𝕂 and the equations (1) and (2) are still satisfied for the residue classes.

The ideals I and J are biideals. In fact, let x ∈ I then ⟨Δ(x), a ⊗ b⟩ = ⟨x, ab⟩ = 0 hence Δ(x) ∈ Ker(φ ⊗ φ : L ⊗ L ⟶ Hom(H ⊗ H, 𝕂) = I ⊗ L + L ⊗ I (the last equality is an easy exercise in linear algebra) and ε(x) = ⟨x, 1⟩ = 0. Hence as in the proof of Theorem 2.6.3 we get that \overline{L} = L/I and \overline{H} = H/J are bialgebras. Since ⟨S(x), a⟩ = ⟨x, S(a)⟩ = 0 we have an induced homomorphism \overline{S} : \overline{L} ⟶ \overline{L}. The identities satisfied in L hold also for the residue classes in \overline{L} so that L and similarly \overline{H} become Hopf algebras. Finally we have by definition of I that ⟨\overline{x}, \overline{a}⟩ = ⟨x, a⟩ = 0 for all a ∈ H iff a ∈ I or \overline{a} = 0. Thus the bilinear form \overline{ev} : \overline{L} ⊗ \overline{H} ⟶ K defines a duality.

Problem 2.7.1. (in Linear Algebra)

1. For U ⊆ V define U^{⊥} := {f ∈ V^{∗}|f(U) = 0}. For Z ⊆ V^{∗} define Z^{⊥} := {v ∈ V |Z(v) = 0}. Show that the following hold:

(a) U ⊆ V ⟹ U = U^{⊥⊥};

(b) Z ⊆ V^{∗} and dim Z < ∞ ⟹ Z = Z^{⊥⊥};

(c) {U ⊆ V | dim V/U < ∞} ≅ {Z ⊆ V^{∗}| dim Z < ∞} under the maps

U ↦ U^{⊥} and Z ↦ Z^{⊥}.

2. Let V = ⨁_{i=1}^{∞} 𝕂x_{i} be an infinite-dimensional vector space. Find an element g ∈ (V ⊗ V)^{∗} that is not in V^{∗} ⊗ V^{∗} (⊆ (V ⊗ V)^{∗}).

Definition 2.7.7. Let A be an algebra. We define A^{o} := {f ∈ A^{∗}|∃ ideal _{A}I_{A} ⊆ A : dim(A/I) < ∞ and f(I) = 0}.

Lemma 2.7.8. Let A be an algebra and f ∈ A^{∗}. The following are equivalent:

1. f ∈ A^{o};

2. there exists I_{A} ⊆ A such that dim A/I < ∞ and f(I) = 0;

3. A · f ⊆ _{A} Hom_{𝕂}(.A_{A}, .𝕂) is finite dimensional;

4. A · f · A is finite dimensional;

5. ∇^{∗}(f) ∈ A^{∗} ⊗ A^{∗}.

Proof. 1. ⟹ 2. and 4. ⟹ 3. are trivial.

2. ⟹ 3. Let I_{A} ⊆ A with f(I) = 0 and dim A/I < ∞. Write A^{∗} ⊗ A ⟶ 𝕂 as ⟨g, a⟩. Then ⟨af, i⟩ = ⟨f, ia⟩ = 0 hence Af ⊂ I^{⊥} and dim Af < ∞.

3. ⟹ 2. Let dim Af < ∞. Then I_{A} := (Af)^{⊥} is an ideal of finite codimension in A and f(I) = 0 holds.

2. ⟹ 1. Let I_{A} ⊂ A with dim A/I_{A} < ∞ and f(I) = 0 be given. Then right multiplication induces φ : A ⟶ Hom_{𝕂}(A/I., A/I.) and dim End_{𝕂}(A/I) < ∞. Thus J = Ker(φ) ⊆ A is a two sided ideal of finite codimension and J ⊂ I (since φ(j)(¯1) = 0 = ¯1 · j = j̄ implies j ∈ I). Furthermore we have f(J) ⊆ f(I) = 0.

1. ⟹ 4. ⟨afb, i⟩ = ⟨f, bia⟩ = 0 implies A · f · A ⊆ _{A}I_{A}^{⊥} hence dim AfA < ∞.

3. ⟹ 5. We observe that ∇^{∗}(f) = f∇ ∈ (A ⊗ A)^{∗}. We want to show that ∇^{∗}(f) ∈ A^{∗} ⊗ A^{∗}. Let g_{1}, …, g_{n} be a basis of Af. Then there exist h_{1}, …, h_{n} ∈ A^{∗} such that bf = ∑ h_{i}(b)g_{i}. Let a, b ∈ A. Then ⟨∇^{∗}(f), a ⊗ b⟩ = ⟨f, ab⟩ = ⟨bf, a⟩ = ∑ h_{i}(b)g_{i}(a) = ⟨∑ g_{i} ⊗ h_{i}, a ⊗ b⟩ so that ∇^{∗}(f) = ∑ g_{i} ⊗ h_{i} ∈ A^{∗} ⊗ A^{∗}.

5. ⟹ 3. Let ∇^{∗}(f) = ∑ g_{i} ⊗ h_{i} ∈ A^{∗} ⊗ A^{∗}. Then bf = ∑ h_{i}(b)g_{i} for all b ∈ A as before. Thus Af is generated by the g_{1}, …, g_{n}.

Proposition 2.7.9. Let (A, m, u) be an algebra. Then we have m^{∗}(A^{o}) ⊆ A^{o}⊗A^{o}. Furthermore (A^{o}, Δ, ε) is a coalgebra with Δ = m^{∗} and ε = u^{∗}.

Proof. Let f ∈ A^{o} and let g_{1}, …, g_{n} be a basis for Af. Then we have m^{∗}(f) = ∑ g_{i}⊗h_{i} for suitable h_{i} ∈ A^{∗} as in the proof of the previous proposition. Since g_{i} ∈ Af we get Ag_{i} ⊆ Af and dim(Ag_{i}) < ∞ and hence g_{i} ∈ A^{o}. Choose a_{1}, …, a_{n} ∈ A such that g_{i}(a_{j}) = δ_{ij}. Then (fa_{j})(a) = f(a_{j}a) = ⟨m^{∗}(f), a_{j} ⊗ a⟩ = ∑ g_{i}(a_{j})h_{i}(a) = h_{j}(a) implies fa_{j} = h_{j} ∈ fA. Observe that dim(fA) < ∞ hence dim(h_{j}A) < ∞, so that h_{j} ∈ A^{o}. This proves m^{∗}(f) ∈ A^{o} ⊗ A^{o}.

One checks easily that counit law and coassociativity hold.

Theorem 2.7.10. (The Sweedler dual:) Let (B, m, u, Δ, ε) be a bialgebra. Then (B^{o}, Δ^{∗}, ε^{∗}, m^{∗}, u^{∗}) again is a bialgebra. If B = H is a Hopf algebra with antipode S, then S^{∗} is an antipode for B^{o} = H^{o}.

Proof. We know that (B^{∗}, Δ^{∗}, ε^{∗}) is an algebra and that (B^{o}, m^{∗}, u^{∗}) is a coalgebra. We show now that B^{o} ⊆ B^{∗} is a subalgebra. Let f, g ∈ B^{o} with dim(Bf) < ∞ and dim(Bg) < ∞. Let a ∈ B. Then we have (a(fg))(b) = (fg)(ba) = ∑ f(b_{(1)}a_{(1)})g(b_{(2)}a_{(2)}) = ∑(a_{(1)}f)(b_{(1)})(a_{(2)}g)(b_{(2)}) = ∑((a_{(1)}f)(a_{(2)}g))(b) hence a(fg) = ∑(a_{(1)}f)(a_{(2)}g) ∈ (Bf)(Bg). Since dim(Bf)(Bg) < ∞ we have dim(B(fg)) < ∞ so that fg ∈ B^{o}. Furthermore we have ε ∈ B^{o}, since Ker(ε) has codimension 1. Thus B^{o} ⊆ B^{∗} is a subalgebra. It is now easy to see that B^{o} is a bialgebra.

Now let S be the antipode of H. We show S^{∗}(H^{o}) ⊆ H^{o}. Let a ∈ H, f ∈ H^{o}. Then ⟨aS^{∗}(f), b⟩ = ⟨S^{∗}(f), ba⟩ = ⟨f, S(ba)⟩ = ⟨f, S(a)S(b)⟩ = ⟨fS(a), S(b)⟩ = ⟨S^{∗}(fS(a)), b⟩. This implies aS^{∗}(f) = S^{∗}(fS(a)) and HS^{∗}(f) = S^{∗}(fS(H)) ⊆ S^{∗}(fH). Since f ∈ H^{o} we get dim(fH) < ∞ so that dim(S^{∗}(fH)) < ∞ and dim(HS^{∗}(f)) < ∞. This shows S^{∗}(f) ∈ H^{o}. The rest of the proof is now trivial.

Definition 2.7.11. Let G = 𝕂-cAlg(H, -) be an affine group and R ∈ 𝕂-cAlg. We define G⊗_{𝕂}R := G|_{R-cAlg} to be the restriction to commutative R-algebras. The functor G ⊗_{𝕂} R is represented by H ⊗ R ∈ R-cAlg:

G|_{R-cAlg}(A) = 𝕂-cAlg(H, A) ≅ R-cAlg(H ⊗ R, A).

Theorem 2.7.12. (The Cartier dual:) Let H be a finite dimensional commutative cocommutative Hopf algebra. Let G = 𝕂-cAlg(H, -) be the associated affine group and let D(G) := 𝕂-cAlg(H^{∗}, -) be the dual group. Then we have

D(G) = Gr(G, G_{m})

where Gr(G, G_{m})(R) = Gr(G ⊗_{𝕂} R, G_{m} ⊗_{𝕂} R) is the set of group (-functor) homomorphisms and G_{m} is the multiplicative group.

Proof. We have Gr(G, G_{m})(R) = Gr(G⊗_{𝕂}R, G_{M}⊗_{𝕂}R) ≅ R-Hopf-Alg(𝕂[t, t^{−1}]⊗ R, H ⊗R) ≅ R-Hopf-Alg(R[t, t^{−1}], H ⊗R) ≅ {x ∈ U(H ⊗R)|Δ(x) = x⊗x, ε(x) = 1}, since Δ(x) = x ⊗ x and ε(x) = 1 imply xS(x) = ε(x) = 1.

Consider x ∈ Hom_{R}((H ⊗ R)^{∗}, R) = Hom_{R}(H^{∗} ⊗ R, R). Then Δ(x) = x ⊗ x iff x(v^{∗}w^{∗}) = ⟨x, v^{∗}w^{∗}⟩ = ⟨Δ(x), v^{∗} ⊗ w^{∗}⟩ = x(v^{∗})x(w^{∗}) and ε(x) = 1 iff ⟨x, ε⟩ = 1. Hence x ∈ R-cAlg((H ⊗ R)^{∗}, R) ≅ 𝕂-cAlg(H^{∗}, R) = D(G)(R).