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CHAPTER 4 The Infinitesimal Theory

1. Integrals and Fourier Transforms

Assume for this chapter that 𝕂 is a field.

Lemma 4.1.1. Let C be a finite dimensional coalgebra. Every right C-comodule M is a left C^{∗}-module by c^{∗}m = ∑ m_{(M)}⟨c^{∗}, m_{(1)}⟩ and conversely by δ(m) = ∑_{i} c_{i}^{∗}m⊗ c_{i} where ∑ c_{i}^{∗} ⊗ c_{i} is the dual basis.

Proof. We check that M becomes a left C^{∗}-module

\begin{aligned}(c^{∗}c′^{∗})m & = ∑ m_{(M)}⟨c^{∗}c′^{∗}, m_{(1)}⟩ = ∑ m_{(M)}⟨c^{∗}, m_{(1)}⟩⟨c′^{∗}, m_{(2)}⟩ \\ & = c^{∗} ∑ m_{(M)}⟨c′^{∗}, m_{(1)}⟩ = c^{∗}(c′^{∗}m). \end{aligned}

It is easy to check that the two constructions are inverses of each other. In particular assume that M is a right C-comodule. Choose m_{i} such that δ(m) = ∑ m_{i} ⊗ c_{i}. Then c_{j}^{∗}m = ∑ m_{i}⟨c_{j}^{∗}, c_{i}⟩ = m_{j} and ∑ c_{i}^{∗}m ⊗ c_{i} = ∑ m_{i} ⊗ c_{i} = δ(m).

Definition 4.1.2. 1. Let A be an algebra with augmentation ε : A ⟶ 𝕂, an algebra homomorphism. Let M be a left A-module. Then ^{A}M = {m ∈ M|am = ε(a)m} is called the space of left invariants of M.

This defines a functor ^{A}- : A − Mod ⟶ Vec.

2. Let C be a coalgebra with a group-like element 1 ∈ C. Let M be a right C-comodule. Then M^{coC} := {m ∈ M|δ(m) = m ⊗ 1} is called the space of right coinvariants of M.

This defines a functor -^{coC} : Comod-C ⟶ Vec.

Lemma 4.1.3. Let C be a finite dimensional coalgebra with a group like element 1 ∈ C. Then A := C^{∗} is an augmented algebra with augmentation ε : C^{∗} ∋ a ↦ ⟨a, 1⟩ ∈ 𝕂. Let M be a right C-comodule. Then M is a left C^{∗}-module and we have

C^{∗}M = M^{coC}.

Proof. Since 1 ∈ C is group-like we have ε_{A}(ab) = ⟨ab, 1⟩ = ⟨a, 1⟩⟨b, 1⟩ = ε_{A}(a)ε_{A}(b) and ε_{A}(1_{A}) = ⟨1_{A}, 1_{C}⟩ = ε_{C}(1_{C}) = 1.

We have m ∈ M^{coH} iff δ(m) = ∑ m_{(M)} ⊗ m_{(1)} = m ⊗ 1 iff ∑ m_{(M)}⟨a, m_{(1)}⟩ = m⟨a, 1⟩ for all a ∈ A = C^{∗} and by identifying C^{∗}⊗C = Hom(C^{∗}, C^{∗}) iff am = ε_{A}(a)m iff m ∈ ^{A}M.

Remark 4.1.4. The theory of Fourier transforms contains the following statements. Let H be the (Schwartz) space of infinitely differentiable functions f : ℝ ⟶ ℂ, such that f and all derivatives rapidly decrease at infinity. (f decreases rapidly at infinity if |x|^{m}f(x) is bounded for all m.) This space is an algebra (without unit) under the multiplication of values. There is a second multiplication on H, the convolution

(f ∗ g)(x) = (2π)^{−1/2} ∫_{−∞}^{+∞} f(t)g(x − t)dt.

The Fourier transform is a homomorphism ˆ- : H ⟶ H defined by

f̂(x) = (2π)^{−1/2} ∫_{−∞}^{+∞} f(t)e^{−itx}dt.

It satisfies the identity (f ∗ g)ˆ = f̂ĝ hence it is an algebra homomorphism. We want to find an analogue of this theory for finite quantum groups.

A similar example is the following. Let G be a locally compact topological group. Let μ be the (left) Haar measure on G and ∫ f := ∫_{G} f(x)dμ(x) be the Haar integral.

The Haar measure is left invariant in the sense that μ(E) = μ(gE) for all g ∈ G and all compact subsets E of G. The Haar measure exists and is unique up to a positive factor. The Haar integral is translation invariant i.e. for all y ∈ G we have ∫ f(yx)dμ(x) = ∫ f(x)dμ(x).

If μ is a left-invariant Haar measure then there is a continuous homomorphism mod : G ⟶ (ℝ^{+}, ·) such that ∫ f(xy^{−1})dμ(x) = mod (y) ∫ (f(x)dμ(x). The homomorphism μ does not depend on f and is called the modulus of G. The group G is called unimodular if the homomorphism mod is the identity.

If G is a compact, or discrete, or Abelian group, or a connected semisimple or nilpotent Lie group, then G is unimodular.

Let G be a quantum group (or a quantum monoid) with function algebra H an arbitrary Hopf algebra. We also use the algebra of linear functionals H^{∗} = Hom(H, 𝕂) (called the bialgebra of G in the French literature). The operation H^{∗} ⊗ H ∋ a ⊗ f ↦ ⟨a, f⟩ ∈ 𝕂 is nondegenerate on both sides. We denote the elements of H by f, g, h ∈ H, the elements of H^{∗} by a, b, c ∈ H^{∗}, the (non existing) elements of the quantum group G by x, y, z ∈ G.

Remark 4.1.5. In 2.4.8 we have seen that the dual vector space H^{∗} of a finite dimensional Hopf algebra H is again a Hopf algebra. The Hopf algebra structures are connected by the evaluation bilinear form

ev : H^{∗} ⊗ H ∋ a ⊗ f ↦ ⟨a, f⟩ ∈ 𝕂

as follows:

\begin{matrix}⟨a ⊗ b, ∑ f_{(1)} ⊗ f_{(2)}⟩ = ⟨ab, f⟩, & ⟨∑ a_{(1)} ⊗ a_{(2)}, f ⊗ g⟩ = ⟨a, fg⟩, \\ ⟨a, 1⟩ = ε(a), & ⟨1, f⟩ = ε(f), \end{matrix} ; ⟨a, S(f)⟩ = ⟨S(a), f⟩.

Definition 4.1.6. 1. The linear functionals a ∈ H^{∗} are called generalized integrals on H ([Riesz-Nagy] S.123).

2. An element ∫ ∈ H^{∗} is called a left (invariant) integral on H if

a∫ = ⟨a, 1_{H}⟩∫

or a ∫ = ε_{H^{∗}}(a) ∫ for all a ∈ H^{∗}.

3. An element δ ∈ H is called a left integral in H if

fδ = ε(f)δ

for all f ∈ H.

4. The set of left integrals in H is denoted by Int_{l}(H), the set of right integrals by Int_{r}(H). The set of left (right) integrals on H is Int_{l}(H^{∗}) (Int_{r}(H^{∗})).

5. A Hopf algebra H is called unimodular if Int_{l}(H) = Int_{r}(H).

Lemma 4.1.7. The left integrals Int_{l}(H^{∗}) form a two sided ideal of H^{∗}. If the antipode S is bijective then S induces an isomorphism S : Int_{l}(H^{∗}) ⟶ Int_{r}(H^{∗}).

Proof. For ∫ in Int_{l}(H^{∗}) we have a ∫ = ε(a) ∫ ∈ Int_{l}(H^{∗}) and a ∫ b = ε(a) ∫ b hence ∫ b ∈ Int_{l}(H^{∗}). If S is bijective then the induced map S : H^{∗} ⟶ H^{∗} is also bijective and satisfies S(∫)b = S(∫)S(S^{−1}(b)) = S(S^{−1}(b) ∫) = S(∫)ε(b) hence S(∫) ∈ Int_{r}(H^{∗}).

Remark 4.1.8. Maschke’s Theorem has an extension to finite dimensional Hopf algebras: ε(∫) ≠ 0 iff H^{∗} is semisimple.

Corollary 4.1.9. Let H be a finite dimensional Hopf algebra. Then H^{∗} is a left H^{∗}-module by the usual multiplication, hence a right H-comodule. We have

(H^{∗})^{coH} = Int_{l}(H^{∗}).

Proof. By definition we have Int_{l}(H^{∗}) = ^{H^{∗}}H^{∗}.

Example 4.1.10. Let G be a finite group. Let H := Map(G, 𝕂) be the Hopf algebra defined by the following isomorphism

𝕂^{G} = Map(G, 𝕂) ≅ Hom(𝕂G, 𝕂) = (𝕂G)^{∗}.

This isomorphism between the vector space 𝕂^{G} of all set maps from the group G to the base ring 𝕂 and the dual vector space (𝕂G)^{∗} of the group algebra 𝕂G defines the structure of a Hopf algebra on 𝕂^{G}.

We regard H := 𝕂^{G} as the function algebra on the set G. In the sense of algebraic geometry this is not quite true. The algebra 𝕂^{G} represents a functor from 𝕂-cAlg to Set that has G as value for all connected algebras A in particular for all field extensions of 𝕂.

As before we use the map ev : 𝕂G ⊗ 𝕂^{G} ⟶ 𝕂. The multiplication of 𝕂^{G} is given by pointwise multiplication of maps since ⟨x, ff′⟩ = ⟨∑ x_{(1)} ⊗ x_{(2)}, f ⊗ f′⟩ = ⟨x ⊗ x, f ⊗ f′⟩ = ⟨x, f⟩⟨x, f′⟩ for all f, f′ ∈ 𝕂^{G} and all x ∈ G. The unit element 1_{𝕂^{G}} of 𝕂^{G} is the map ε : 𝕂G ⟶ 𝕂 restricted to G, hence ε(x) = 1 = ⟨x, 1_{𝕂^{G}}⟩ for all x ∈ G. The antipode of f ∈ 𝕂^{G} is given by S(f)(x) = ⟨x, S(f)⟩ = f(x^{−1}).

The elements of the dual basis (x^{∗}|x ∈ G) with ⟨x, y^{∗}⟩ = δ_{x,y} considered as maps from G to 𝕂 form a basis of 𝕂^{G}. They satisfy the conditions

x^{∗}y^{∗} = δ_{x,y}x^{∗} and ∑_{x∈G} x^{∗} = 1_{𝕂^{G}}

since ⟨z, x^{∗}y^{∗}⟩ = ⟨z, x^{∗}⟩⟨z, y^{∗}⟩ = δ_{z,x}δ_{z,y} = δ_{x,y}⟨z, x^{∗}⟩ and ⟨z, ∑_{x∈G} x^{∗}⟩ = 1 = ⟨z, 1_{𝕂^{G}}⟩.

Hence the dual basis (x^{∗}|x ∈ G) is a decomposition of the unit into a set of minimal orthogonal idempotents and the algebra of 𝕂^{G} has the structure

𝕂^{G} = ⊕_{x∈G}𝕂x^{∗} ≅ 𝕂 × … × 𝕂.

In particular 𝕂^{G} is commutative and semisimple.

The diagonal of 𝕂^{G} is

Δ(x^{∗}) = ∑_{y∈G} y^{∗} ⊗ (y^{−1}x)^{∗} = ∑_{y,z∈G,yz=x} y^{∗} ⊗ z^{∗}

since

\begin{matrix}⟨z ⊗ u, Δ(x^{∗})⟩ = ⟨zu, x^{∗}⟩ = δ_{x,zu} = δ_{z^{−1}x,u} = ∑_{y∈G} δ_{y,z}δ_{y^{−1}x,u} \\ = ∑_{y∈G}⟨z, y^{∗}⟩⟨u, (y^{−1}x)^{∗}⟩ = ⟨z ⊗ u, ∑_{y∈G} y^{∗} ⊗ (y^{−1}x)^{∗}⟩. \end{matrix}

Let a ∈ 𝕂G. Then a defines a map ã : G ⟶ 𝕂 ∈ 𝕂^{G} by a = ∑_{x∈G} ã(x)x. For arbitrary f ∈ 𝕂^{G} and a ∈ 𝕂G this gives

⟨a, f⟩ = f(∑_{x∈G} ã(x)x) = ∑_{x∈G} ã(x)f(x).

The counit of 𝕂^{G} is given by ε(x^{∗}) = δ_{x,e} where e ∈ G is the unit element.

The antipode is, as above, S(x^{∗}) = (x^{−1})^{∗}.

We consider H = 𝕂^{G} as the function algebra on the finite group G and 𝕂G as the dual space of H = 𝕂^{G} hence as the set of distributions on H.

Then

∫ := ∑_{x∈G} x ∈ H^{∗} = 𝕂G (1)

is a (two sided) integral on H since ∑_{x∈G} yx = ∑_{x∈G} x = ε(y) ∑_{x∈G} x = ∑_{x∈G} yx. We write

∫ f(x)dx := ⟨∫ , f⟩ = ∑_{x∈G} f(x).

We have seen that there is a decomposition of the unit 1 ∈ 𝕂^{G} into a set of primitive orthogonal idempotents {x^{∗}|x ∈ G} such that every element f ∈ 𝕂^{G} has a unique representation f = ∑ f(x)x^{∗}. Since ∫ y^{∗} = ∑_{x∈G}⟨x, y^{∗}⟩ we get ∫ fy^{∗} = ∑_{x∈G}⟨x, fy^{∗}⟩ = ∑ f(x)y^{∗}(x) = f(y) hence

f = ∑(∫ f(x)y^{∗}(x)dx)y^{∗}.

Problem 4.1.1. Describe the group valued functor 𝕂-cAlg(𝕂^{G}, −) in terms of sets and their group structure.

Definition and Remark 4.1.11. Let 𝕂 be an algebraicly closed field and let G be a finite abelian group (replacing ℝ above). Assume that the characteristic of 𝕂 does not divide the order of G. Let H = 𝕂^{G}. We identify 𝕂^{G} = Hom(𝕂G, 𝕂) along the linear expansion of maps as in Example 2.1.10.

Let us consider the set Ĝ := {χ : G ⟶ 𝕂^{∗}|χ group homomorphism}. Since 𝕂^{∗} is an abelian group, the set Ĝ is an abelian group by pointwise multiplication.

The group Ĝ is called the character group of G.

Obviously the character group is a multiplicative subset of 𝕂^{G} = Hom(𝕂G, 𝕂). Actually it is a subgroup of 𝕂-cAlg(𝕂G, 𝕂) ⊆ Hom(𝕂G, 𝕂) since the elements χ ∈ Ĝ expand to algebra homomorphisms: χ(ab) = χ(∑ α_{x}x ∑ β_{y}y) = ∑ α_{x}β_{y}χ(xy) = χ(a)χ(b) and χ(1) = χ(e) = 1. Conversely an algebra homomorphism f ∈ 𝕂-cAlg(𝕂G, 𝕂) restricts to a character f : G ⟶ 𝕂^{∗}. Thus Ĝ = 𝕂-cAlg(𝕂G, 𝕂), the set of rational points of the affine algebraic group represented by 𝕂G.

There is a more general observation behind this remark.

Lemma 4.1.12. Let H be a finite dimensional Hopf algebra. Then the set Gr(H^{∗}) of group like elements of H^{∗} is equal to 𝕂-Alg(H, 𝕂).

Proof. In fact f : H ⟶ 𝕂 is an algebra homomorphism iff ⟨f ⊗ f, a ⊗ b⟩ = ⟨f, a⟩⟨f, b⟩ = ⟨f, ab⟩ = ⟨Δ(f), a ⊗ b⟩ and 1 = ⟨f, 1⟩ = ε(f).

Hence there is a Hopf algebra homomorphism φ : 𝕂Ĝ ⟶ 𝕂^{G} by 2.1.5.

Proposition 4.1.13. The Hopf algebra homomorphism φ : 𝕂Ĝ ⟶ 𝕂^{G} is bijective.

Proof. We give the proof by several lemmas.

Lemma 4.1.14. Any set of group like elements in a Hopf algebra H is linearly independent.

Proof. Assume there is a linearly dependent set {x_{0}, x_{1}, …, x_{n}} of group like elements in H. Choose such a set with n minimal. Obviously n ≥ 1 since all elements are non zero. Thus x_{0} = ∑_{i=1}^{n} α_{i}x_{i} and {x_{1}, …, x_{n}} linearly independent. We get

∑_{i,j} α_{i}α_{j}x_{i} ⊗ x_{j} = x_{0} ⊗ x_{0} = Δ(x_{0}) = ∑_{i} α_{i}x_{i} ⊗ x_{i}.

Since all α_{i} ≠ 0 and the x_{i} ⊗ x_{j} are linearly independent we get n = 1 and α_{1} = 1 so that x_{0} = x_{1}, a contradiction.

Corollary 4.1.15. (Dedekind’s Lemma) Any set of characters in 𝕂^{G} is linearly independent.

Thus φ : 𝕂Ĝ ⟶ 𝕂^{G} is injective. Now we prove that the map φ : 𝕂Ĝ ⟶ 𝕂^{G} is surjective.

Lemma 4.1.16. (Pontryagin duality) The evaluation Ĝ × G ⟶ 𝕂^{∗} is a nondegenerate bilinear map of abelian groups.

Proof. First we observe that Hom(C_{n}, 𝕂^{∗}) ≅ C_{n} for a cyclic group of order n since 𝕂 has a primitive n-th root of unity (char(𝕂)−̸ |G|).

Since the direct product and the direct sum coincide in Ab we can use the fundamental theorem for finite abelian groups G ≅ C_{n_{1}} × … × C_{n_{t}} to get Hom(G, 𝕂^{∗}) ≅ G for any abelian group G with char(𝕂)−̸ |G|. Thus Ĝ ≅ G and ˆ Ĝ = G. In particular χ(x) = 1 for all x ∈ G iff χ = 1. By the symmetry of the situation we get that the bilinear form ⟨., .⟩ : Ĝ × G ⟶ 𝕂^{∗} is non-degenerate.

Thus |Ĝ| = |G| hence dim(𝕂Ĝ) = dim(𝕂^{G}). This proves Proposition 2.1.13.

Definition 4.1.17. Let H be a Hopf algebra. A 𝕂-module M that is a right H-module by ρ : M ⊗ H ⟶ M and a right H-comodule by δ : M ⟶ M ⊗ H is called a Hopf module if the diagram

[figure: M ⊗ H; H; ρ; M ⊗ H; δ; M ⊗ H ⊗ H ⊗ H; M ⊗ H ⊗ H ⊗ H; 1⊗τ⊗1; δ⊗Δ; ρ⊗∇]

commutes, i.e. if δ(mh) = ∑ m_{(M)}h_{(1)} ⊗ m_{(1)}h_{(2)} holds for all h ∈ H and all m ∈ M.

Observe that H is an Hopf module over itself. Furthermore each module of the form V ⊗ H is a Hopf module by the induced structure. More generally there is a functor Vec ∋ V ↦ V ⊗ H ∈ Hopf-Mod-H.

Proposition 4.1.18. The two functors -^{coH} : Hopf-Mod-H ⟶ Vec and - ⊗ H : Vec ∋ V ↦ V ⊗ H ∈ Hopf-Mod-H are inverse equivalences of each other.

Proof. Define natural isomorphisms

α : M^{coH} ⊗ H ∋ m ⊗ h ↦ mh ∈ M

with inverse map

α^{−1} : M ∋ m ↦ ∑ m_{(M)}S(m_{(1)}) ⊗ m_{(2)} ∈ M^{coH} ⊗ H

and

β : V ∋ v ↦ v ⊗ 1 ∈ (V ⊗ H)^{coH}

with inverse map

(V ⊗ H)^{coH} ∋ v ⊗ h ↦ vε(h) ∈ V.

Obviously these homomorphisms are natural transformations in M and V. Furthermore α is a homomorphism of H-modules. α^{−1} is well-defined since

δ(∑ m_{(M)}S(m(1))) = ∑ m_{(M)}S(m_{(3)}) ⊗ m_{(1)}S(m_{(2)})

(since M is a Hopf module) = ∑ m_{(M)}S(m_{(2)}) ⊗ ηε(m_{(1)}) = ∑ m_{(M)}S(m_{(1)}) ⊗ 1

hence ∑ m_{(M)}S(m_{(1)}) ∈ M^{coH}. Furthermore α^{−1} is a homomorphism of comodules since

\begin{aligned}δα^{−1}(m) & = δ(∑ m_{(M)}S(m_{(1)}) ⊗ m_{(2)}) = ∑ m_{(M)}S(m_{(1)}) ⊗ m_{(2)} ⊗ m_{(3)} \\ & = ∑ α^{−1}(m_{(M)}) ⊗ m_{(1)} = (α^{−1} ⊗ 1)δ(m). \end{aligned}

Finally α and α^{−1} are inverse to each other by

αα^{−1}(m) = α(∑ m_{(M)}S(m_{(1)}) ⊗ m_{(2)}) = ∑ m_{(M)}S(m_{(1)})m_{(2)} = m

and

\begin{aligned}α^{−1}α(m ⊗ h) & = α^{−1}(mh) = ∑ m_{(M)}h_{(1)}S(m_{(1)}h_{(2)}) ⊗ m_{(2)}h_{(3)} \\ & = ∑ mh_{(1)}S(h_{(2)}) ⊗ h_{(3)} ( by δ(m) = m ⊗ 1 ) = m ⊗ h. \end{aligned}

Thus α and α^{−1} are mutually inverse homomorphisms of Hopf modules.

The image of β is in (V ⊗ H)^{coH} by δ(v ⊗ 1) = v ⊗ Δ(1) = (v ⊗ 1) ⊗ 1. Both β and β^{−1} are 𝕂-linear maps. Furthermore we have

β^{−1}β(v) = β^{−1}(v ⊗ 1) = vε(1) = v

and

\begin{aligned}ββ^{−1} & (∑ v_{i} ⊗ h_{i}) = β(∑ v_{i}ε(h_{i})) = ∑ v_{i}ε(h_{i}) ⊗ 1 = ∑ v_{i} ⊗ ε(h_{i})1 \\ & = ∑ v_{i} ⊗ ε(h_{i(1)})h_{i(2)} ( since ∑ v_{i} ⊗ h_{i} ∈ (V ⊗ H)^{coH} ) = ∑ v_{i} ⊗ h_{i}. \end{aligned}

Thus β and β^{−1} are mutually inverse homomorphisms.

Since H^{∗} = Hom(H, 𝕂) and S : H ⟶ H is an algebra antihomomorphism, the dual H^{∗} is an H-module in four different ways:

\begin{matrix}⟨(f ⇀ a), g⟩ := ⟨a, gf⟩, & ⟨(a ↼ f), g⟩ := ⟨a, fg⟩, \\ ⟨(f ⇁ a), g⟩ := ⟨a, S(f)g⟩, & ⟨(a ↽ f), g⟩ := ⟨a, gS(f)⟩. \end{matrix} (2)

If H is finite dimensional then H^{∗} is a Hopf algebra. The equality ⟨(f ⇀ a), g⟩ = ⟨a, gf⟩ = ∑⟨a_{(1)}, g⟩⟨a_{(2)}, f⟩ implies

(f ⇀ a) = ∑ a_{(1)}⟨a_{(2)}, f⟩. (3)

Analogously we have

(a ↼ f) = ∑⟨a_{(1)}, f⟩a_{(2)}. (4)

Proposition 4.1.19. Let H be a finite dimensional Hopf algebra. Then H^{∗} is a right Hopf module over H.

Proof. H^{∗} is a left H^{∗}-module by left multiplication hence by 2.1.1 a right Hcomodule by δ(a) = ∑_{i} b_{i}^{∗}a⊗b_{i}. Let f, g ∈ H and a, b ∈ H^{∗}. The (left) multiplication of H^{∗} satisfies

ab = ∑ b_{(H^{∗})}⟨a, b_{(1)}⟩.

We use the right H-module structure

(a ↽ f) = ∑ a_{(1)}⟨S(f), a_{(2)}⟩.

on H^{∗} = Hom(H, 𝕂).

Now we check the Hopf module property. Let a, b ∈ H^{∗} and f, g ∈ H. We apply H^{∗} ⊗ H to its dual H ⊗ H^{∗} and get

δ(a ↽f)(g ⊗ b) = ∑⟨(a ↽ f)_{(H^{∗})}, g⟩⟨b, (a ↽ f)_{(1)}⟩ = ⟨b(a ↽ f), g⟩
= ∑⟨b, g_{(1)}⟩⟨(a ↽ f), g_{(2)}⟩ = ∑⟨b, g_{(1)}⟩⟨a, g_{(2)}S(f)⟩

= ∑⟨b, g_{(1)}ε(f_{(2)})⟩⟨a, g_{(2)}S(f_{(1)})⟩ = ∑⟨(f_{(3)} ⇀ b), g_{(1)}S(f_{(2)})⟩⟨a, g_{(2)}S(f_{(1)})⟩

= ∑⟨(f_{(2)} ⇀ b)a, gS(f_{(1)})⟩ = ∑⟨((f_{(2)} ⇀ b)a) ↽ f_{(1)}, g⟩
= ∑⟨(a_{(H^{∗})}⟨(f_{(2)} ⇀ b), a_{(1)}⟩) ↽ f_{(1)}, g⟩
= ∑⟨(a_{(H^{∗})} ↽ f_{(1)})⟨(f_{(2)} ⇀ b), a_{(1)}⟩, g⟩ = ∑⟨(a_{(H^{∗})} ↽ f_{(1)})⟨b, a_{(1)}f_{(2)}⟩, g⟩

hence δ(a ↽ f) = ∑(a_{(H^{∗})} ↽ f_{(1)}) ⊗ a_{(1)}f_{(2)}.

Theorem 4.1.20. Let H be a finite dimensional Hopf algebra. Then the antipode S is bijective, the space of left integrals Int_{l}(H^{∗}) has dimension 1, and the homomorphism

H ∋ f ↦ (f ⇀ ∫) = ∑_{∫(1)}⟨_{∫(2)}, f⟩ ∋ H^{∗}

is bijective for any 0 ≠ ∫ ∈ Int_{l}(H^{∗}).

Proof. By Proposition 2.1.19 H^{∗} is a right Hopf module over H. By Proposition 2.1.18 there is an isomorphism α : (H^{∗})^{coH} ⊗ H ∋ a ⊗ f ↦ (a ↽ f) = (S(f) ⇀ a) ∈ H^{∗}. Since (H^{∗})^{coH} ≅ Int_{l}(H^{∗}) by 2.1.9 we get

Int_{l}(H^{∗}) ⊗ H ≅ H^{∗}

as right H-Hopf modules by the given map. This shows dim(Int_{l}(H^{∗})) = 1. So we get an isomorphism H ∋ f ↦ (∫ ↽ f) ∈ H^{∗} that is a composition of S and f ↦ (f ⇀ ∫). Since H is finite dimensional both of these maps are bijective.

If G is a finite group then every generalized integral a ∈ 𝕂G can be written with a uniquely determined g ∈ H = 𝕂^{G} as

⟨a, f⟩ = ∫ f(x)S(g)(x)dx = ∑_{x∈G} f(x)g(x^{−1}) (5)

for all f ∈ H.

If G is a finite Abelian group then each group element (rational integral) y ∈ G ⊆ 𝕂G can be written as

y = ∑_{x∈G} ∑_{χ∈Ĝ} β_{χ}⟨x^{−1}, χ⟩x

since

\begin{aligned}⟨y, f⟩ & = ⟨(∫ ↽ ∑_{χ∈Ĝ} β_{χ}χ), f⟩ = ⟨∫ , fS(∑_{χ∈Ĝ} β_{χ}χ)⟩ \\ & = ∑_{x∈G}⟨x, f⟩ ∑_{χ∈Ĝ} β_{χ}⟨x, S(χ)⟩ = ⟨∑_{x∈G} ∑_{χ∈Ĝ} β_{χ}⟨x^{−1}, χ⟩x, f⟩. \end{aligned}

In particular the matrix (⟨x^{−1}, χ⟩) is invertible.

Let H be finite dimensional. Since ⟨∫ , fg⟩ = ⟨(∫ ↼ f), g⟩ as a functional on g is a generalized integral, there is a unique ν(f) ∈ H such that

⟨∫ , fg⟩ = ⟨∫ , gν(f)⟩ (6)

or

∫ f(x)g(x)dx = ∫ g(x)ν(f)(x)dx. (7)

Although the functions f, g ∈ H of the quantum group do not commute under multiplication, there is a simple commutation rule if the product is integrated.

Proposition and Definition 4.1.21. The map ν : H ⟶ H is an algebra automorphism, called the Nakayama automorphism.

Proof. It is clear that ν is a linear map. We have ∫ fν(gh) = ∫ ghf = ∫ hfν(g) = ∫ fν(g)ν(h) hence ν(gh) = ν(g)ν(h) and ∫ fν(1) = ∫ f hence ν(1) = 1. Furthermore if ν(g) = 0 then 0 = ⟨∫ , fν(g)⟩ = ⟨∫ , gf⟩ = ⟨(f ⇀ ∫), g⟩ for all f ∈ H hence ⟨a, g⟩ = 0 for all a ∈ H^{∗} hence g = 0. So ν is injective hence bijective.

Corollary 4.1.22. The map H ∋ f ↦ (∫ ↼ f) ∈ H^{∗} is an isomorphism.

Proof. We have

(∫ ↼ f) = (ν(f) ⇀ ∫)

since ⟨(∫ ↼ f), g⟩ = ⟨∫ , fg⟩ = ⟨∫ , gν(f)⟩ = ⟨(ν(f) ⇀ ∫), g⟩. This implies the corollary.

If G is a finite group and H = 𝕂^{G} then H is commutative hence ν = id.

Definition 4.1.23. An element δ ∈ H is called a Dirac δ-function if δ is a left invariant integral in H with ⟨∫ , δ⟩ = 1, i.e. if δ satisfies

fδ = ε(f)δ  and  ∫ δ(x)dx = 1

for all f ∈ H. If H has a Dirac δ-function then we write

∫ ^{∗} a(x)dx = ∫^{∗}a := ⟨a, δ⟩. (8)

Proposition 4.1.24..

1. If H is finite dimensional then there exists a unique Dirac δ-function δ.

2. If H is infinite dimensional then there exists no Dirac δ-function.

Proof. 1. Since H ∋ f ↦ (f ⇀ ∫) ∈ H^{∗} is an isomorphism there is a δ ∈ H such that (δ ⇀ ∫) = ε. Then (fδ ⇀ ∫) = (f ⇀ (δ ⇀ ∫)) = (f ⇀ ε) = ε(f)ε = ε(f)(δ ⇀ ∫) which implies fδ = ε(f)δ. Furthermore we have ⟨∫ , δ⟩ = ⟨∫ , 1_{H}δ⟩ = ⟨(δ ⇀ ∫), 1_{H}⟩ = ε(1_{H}) = 1_{𝕂}.

2. is [Sweedler] exercise V.4.

Lemma 4.1.25. Let H be a finite dimensional Hopf algebra. Then ∫ ∈ H^{∗} is a left integral iff

a(∑ ∫_{(1)} ⊗ S(∫_{(2)})) = (∑ ∫_{(1)} ⊗ S(∫_{(2)}))a (9)

iff

∑ S(a)∫_{(1)} ⊗ ∫_{(2)} = ∑ ∫_{(1)} ⊗ a∫_{(2)} (10)

iff

∑ f_{(1)}⟨∫ , f_{(2)}⟩ = ⟨∫ , f⟩1_{H}. (11)

Proof. Let ∫ be a left integral. Then

∑ a_{(1)}∫_{(1)} ⊗ S(∫_{(2)})S(a_{(2)}) = ∑(a∫)_{(1)} ⊗ S((a∫)_{(2)}) = ε(a)(∑ ∫_{(1)} ⊗ S(∫_{(2)}))

for all a ∈ H. Hence

(∑ ∫_{(1)} ⊗ S(∫_{(2)}))a = ∑ ε(a_{(1)})(∫_{(1)} ⊗ S(∫_{(2)}))a_{(2)}
= ∑ a_{(1)}∫_{(1)} ⊗ S(∫_{(2)})S(a_{(2)})a_{(3)}
= ∑ a_{(1)}∫_{(1)} ⊗ S(∫_{(2)})ε(a_{(2)}) = a(∑ ∫_{(1)} ⊗ S(∫_{(2)})).

Conversely a(∑ ∫_{(1)} ε(S(∫_{(2)}))) = (∑ ∫_{(1)} ε(S(∫_{(2)})a)) = ε(a)(∑ ∫_{(1)} ε(S(∫_{(2)}))), hence ∫ = ∑ ∫_{(1)} ε(S(∫_{(2)})) is a left integral.

Since S is bijective the following holds

\begin{matrix}∑ S(a) ∫_{(1)} ⊗ ∫_{(2)} = ∑ S(a) ∫_{(1)} ⊗S^{−1}(S(∫_{(2)})) \\ = ∑ ∫_{(1)} ⊗S^{−1}(S(∫_{(2)})S(a)) = ∑ ∫_{(1)} ⊗a ∫_{(2)} . \end{matrix}

The converse follows easily.

If ∫ ∈ Int_{l}(H) is a left integral then ∑⟨a, f_{(1)}⟩⟨∫ , f_{(2)}⟩ = ⟨a ∫ , f⟩ = ⟨a, 1_{H}⟩⟨∫ , f⟩. Conversely if λ ∈ H^{∗} with (11) is given then ⟨aλ, f⟩ = ∑⟨a, f_{(1)}⟩⟨λ, f_{(2)}⟩ =

⟨a, 1_{H}⟩⟨λ, f⟩ hence aλ = ε(a)λ.

If G is a finite group then

δ(x) = \begin{cases}0 if x ≠ e; \\ 1 if x = e. \end{cases} (12)

In fact since δ is left invariant we get f(x)δ(x) = f(e)δ(x) for all x ∈ G and f ∈ 𝕂^{G}. Since G ⊂ H^{∗} = 𝕂G is a basis, we get δ(x) = 0 if x ≠ e. Furthermore ∫ δ(x)dx = ∑_{x∈G} δ(x) = 1 implies δ(e) = 1. So we have δ = e^{∗}.

If G is a finite Abelian group we get δ = α ∑_{χ∈Ĝ} χ for some α ∈ 𝕂. The evaluation gives 1 = ⟨∫ , δ⟩ = α ∑_{x∈G,χ∈Ĝ}⟨χ, x⟩. Now let λ ∈ Ĝ. Then ∑_{χ∈Ĝ}⟨χ, x⟩ = ∑_{χ∈Ĝ}⟨λχ, x⟩ = ⟨λ, x⟩ ∑_{χ∈Ĝ}⟨χ, x⟩. Since for each x ∈ G \ {e} there is a λ such that ⟨λ, x⟩ ≠ 1 and we get

∑_{χ∈Ĝ}⟨χ, x⟩ = |G|δ_{e,x}.

Hence ∑_{x∈G,χ∈Ĝ}⟨χ, x⟩ = |G| = α^{−1} and

δ = |G|^{−1} ∑_{χ∈Ĝ} χ. (13)

Let H be finite dimensional for the rest of this section. In Corollary 1.22 we have seen that the map H ∋ f ↦ (∫ ↼ f) ∈ H^{∗} is an isomorphism. This map will be called the Fourier transform.

Theorem 4.1.26. The Fourier transform H ∋ f ↦ f̃ ∈ H^{∗} is bijective with

f̃ = (∫ ↼ f) = ∑⟨∫_{(1)}, f⟩∫_{(2)} (14)

The inverse Fourier transform is defined by

ã = ∑ S^{−1}(δ_{(1)})⟨a, δ_{(2)}⟩. (15)

Since these maps are inverses of each other the following formulas hold

\begin{matrix}⟨f̃, g⟩ = ∫ f(x)g(x)dx & ⟨a, b̃⟩ = ^{∫ ∗}S^{−1}(a)(x)b(x)dx \\ f = ∑ S^{−1}(δ_{(1)})⟨f̃, δ_{(2)}⟩ & a = ∑⟨∫_{(1)}, ã⟩ ∫_{(2)} . \end{matrix} (16)

Proof. We use the isomorphisms H ⟶ H^{∗} defined by f̂ := f̃ = (∫ ↼ f) = ∑⟨∫_{(1)}, f⟩ ∫_{(2)} and H^{∗} ⟶ H defined by â := (a ⇀ δ) = ∑ δ_{(1)}⟨a, δ_{(2)}⟩. Because of

⟨a, b̂⟩ = ⟨a, (b ⇀ δ)⟩ = ⟨ab, δ⟩ (17)

and

⟨f̃, g⟩ = ⟨(∫ ↼ f), g⟩ = ⟨∫ , fg⟩ (18)

we get for all a ∈ H^{∗} and f ∈ H

\begin{aligned}⟨a, ˆ f̂⟩ & = ⟨af̂, δ⟩ = ∑⟨a, δ_{(1)}⟩⟨f̂, δ_{(2)}⟩ = ∑⟨a, δ_{(1)}⟩⟨∫ , fδ_{(2)}⟩ ( by Lemma 1.25 ) \\ & = ∑⟨a, S(f)δ_{(1)}⟩⟨∫ , δ_{(2)}⟩ = ⟨a, S(f)⟩⟨∫ , δ⟩ = ⟨a, S(f)⟩. \end{aligned}

This gives ˆf̂ = S(f). So the inverse map of H ⟶ H^{∗} with f̂ = (∫ ↼ f) = f̃ is H^{∗} ⟶ H with S^{−1}(â) = ∑ S^{−1}(δ_{(1)})⟨a, δ_{(2)}⟩ = ã. Then the given inversion formulas are clear.

We note for later use ⟨a, b̃⟩ = ⟨a, S^{−1}(b̂)⟩ = ⟨S^{−1}(a), b̂⟩ = ⟨S^{−1}(a)b, δ⟩.

If G is a finite group and H = 𝕂^{G} then

f̃ = ∑_{x∈G} f(x)x.

Since Δ(δ) = ∑_{x∈G} x^{−1}^{∗} ⊗ x^{∗} where the x^{∗} ∈ 𝕂^{G} are the dual basis to the x ∈ G, we get

ã = ∑_{x∈G}⟨a, x^{∗}⟩x^{∗}.

If G is a finite Abelian group then the groups G and Ĝ are isomorphic so the Fourier transform induces a linear automorphism ˜- : 𝕂^{G} ⟶ 𝕂^{G} and we have

ã = |G|^{−1} ∑_{χ∈Ĝ}⟨a, χ⟩χ^{−1}

By substituting the formulas for the integral and the Dirac δ-function (1) and (13) we get

\begin{matrix}f̃ = ∑_{x∈G} f(x)x, & ã = |G|^{−1} ∑_{χ∈Ĝ} a(χ)χ^{−1}, \\ f = |G|^{−1} ∑_{χ∈Ĝ} f̃(χ)χ^{−1}, & a = ∑_{x∈G} ã(x)x. \end{matrix} (19)

This implies

f̃(χ) = ∑_{x∈G} f(x)χ(x) = ∫ f(x)χ(x)dx (20)

with inverse transform

ã(x) = |G|^{−1} ∑_{χ∈Ĝ} χ(a)χ^{−1}(x). (21)

Corollary 4.1.27. The Fourier transforms of the left invariant integrals in H and H^{∗} are

δ̃ = εν^{−1} ∈ H^{∗}  and  ˜∫ = 1 ∈ H. (22)

Proof. We have ⟨δ̃, f⟩ = ⟨∫ , δf⟩ = ⟨∫ , ν^{−1}(f)δ⟩ = εν^{−1}(f)⟨∫ , δ⟩ = εν^{−1}(f) hence δ̃ = εν^{−1}. From ˜1 = (∫ ↼ 1) = ∫ we get ˜∫ = 1.

Proposition 4.1.28. Define a convolution multiplication on H^{∗} by

⟨a ∗ b, f⟩ := ∑⟨a, S^{−1}(δ_{(1)})f⟩⟨b, δ_{(2)}⟩.

1. INTEGRALS AND FOURIER TRANSFORMS 129

Then the following transformation rule holds for f, g ∈ H: f̃g = f̃∗ g̃. (23) In particular H^{∗} with the convolution multiplication is an associative algebra with unit ˜1_{H} = ∫, i.e. ∫ ∗ a = a ∗ ∫ = a. (24)

Proof. Given f, g, h ∈ H^{∗}. Then

⟨f̃g, h⟩ = ⟨∫ , fgh⟩ = ⟨∫ , fS^{−1}(1_{H})gh⟩⟨∫ , δ⟩
= ∑⟨∫ , fS^{−1}(δ_{(1)})gh⟩⟨∫ , δ_{(2)}⟩ = ∑⟨∫ , fS^{−1}(δ_{(1)})h⟩⟨∫ , gδ_{(2)}⟩
= ∑⟨f̃, S^{−1}(δ_{(1)})h⟩⟨g̃, δ_{(2)}⟩ = ⟨f̃∗ g̃, h⟩.

From (22) we get ˜1_{H} = ∫. So we have f̃ = ˜1f = ˜1 ∗ f̃ = ∫ ∗f̃.

If G is a finite Abelian group and a, b ∈ H^{∗} = 𝕂Ĝ. Then

(a ∗ b)(μ) = |G|^{−1} ∑_{χ,λ∈Ĝ,χλ=μ} a(λ)b(χ).

In fact we have

\begin{aligned}(a ∗ b)(μ) & = ⟨a ∗ b, μ⟩ = ∑⟨a, S^{−1}(δ_{(1)})μ⟩⟨b, δ_{(2)}⟩ \\ & = |G|^{−1} ∑_{χ∈Ĝ}⟨a, χ^{−1}μ⟩⟨b, χ⟩ = |G|^{−1} ∑_{χ,λ∈Ĝ,χλ=μ} a(λ)b(χ). \end{aligned}

One of the most important formulas for Fourier transforms is the Plancherel formula on the invariance of the inner product under Fourier transforms. We have

Theorem 4.1.29. (The Plancherel formula) ⟨a, f⟩ = ⟨f̃, ν(ã)⟩. (25)

Proof. First we have from (16)

⟨a, f⟩ = ∑⟨∫_{(1)}, ã⟩⟨∫_{(2)}, S^{−1}(δ_{(1)})⟩⟨f̃, δ_{(2)}⟩ = ∑⟨∫ , ãS^{−1}(δ_{(1)})⟩⟨f̃, δ_{(2)}⟩
= ∑⟨∫ , S^{−1}(δ_{(1)})ν(ã)⟩⟨f̃, δ_{(2)}⟩ = ∑⟨∫ , S^{−1}(S(ν(ã))δ_{(1)})⟩⟨f̃, δ_{(2)}⟩
= ∑⟨∫ , S^{−1}(δ_{(1)})⟩⟨f̃, ν(ã)δ_{(2)}⟩ = ∑⟨∫ , S^{−1}(δ)_{(2)}⟩⟨f̃, ν(ã)S(S^{−1}(δ)_{(1)})⟩
= ⟨∫ , S^{−1}(δ)⟩⟨f̃, ν(ã)⟩.

Apply this to ⟨∫ , δ⟩. Then we get

1 = ⟨∫ , δ⟩ = ⟨∫ , S^{−1}(δ)⟩⟨δ̃, ν(˜∫)⟩ = ⟨∫ , S^{−1}(δ)⟩εν^{−1}ν(1) = ⟨∫ , S^{−1}(δ)⟩. Hence we get ⟨a, f⟩ = ⟨f̃, ν(ã)⟩.

Corollary 4.1.30. If H is unimodular then ν = S^{2}.

Proof. H unimodular means that δ is left and right invariant. Thus we get

⟨a, f⟩ = ∑⟨∫_{(1)}, ã⟩⟨∫_{(2)}, S^{−1}(δ_{(1)})⟩⟨f̃, δ_{(2)}⟩
= ∑⟨∫ , ãS^{−1}(δ_{(1)})⟩⟨f̃, δ_{(2)}⟩ = ∑⟨∫ , S^{−1}(δ_{(1)}S(ã))⟩⟨f̃, δ_{(2)}⟩
= ∑⟨∫ , S^{−1}(δ_{(1)})⟩⟨f̃, δ_{(2)}S^{2}(ã)⟩ ( since δ is right invariant)
= ⟨∫ , S^{−1}(δ)⟩⟨f̃, S^{2}(ã)⟩ = ⟨f̃, S^{2}(ã)⟩.

Hence S^{2} = ν.

We also get a special representation of the inner product H^{∗} ⊗ H ⟶ 𝕂 by both integrals:

Corollary 4.1.31.

⟨a, f⟩ = ∫ ã(x)f(x)dx = ∫ ^{∗} S^{−1}(a)(x)f̃(x)dx. (26)

Proof. We have the rules for the Fourier transform. From (18) we get ⟨a, f⟩ = ⟨∫ , ãf⟩ = ∫ ã(x)f(x)dx and from (17) ⟨a, f⟩ = ⟨S^{−1}(a)f̃, δ⟩ = ∫^{∗} S^{−1}(a)(x)f̃(x)dx.

The Fourier transform leads to an interesting integral transform on H by double application.

Proposition 4.1.32. The double transform f̆ := (δ ↼ (∫ ↼ f)) defines an automorphism H ⟶ H with

f̆(y) = ∫ f(x)δ(xy)dx.

Proof. We have

⟨y, f̆⟩ = ⟨y, (δ ↼ (∫ ↼ f))⟩ = ⟨(∫ ↼ f)y, δ⟩
= ∑⟨(∫ ↼ f), δ_{(1)}⟩⟨y, δ_{(2)}⟩ = ∑⟨∫ , fδ_{(1)}⟩⟨y, δ_{(2)}⟩
= ∑⟨∫_{(1)}, f⟩⟨∫_{(2)}, δ_{(1)}⟩⟨y, δ_{(2)}⟩ = ∑⟨∫_{(1)}, f⟩⟨∫_{(2)} y, δ⟩
= ∑⟨∫_{(1)}, f⟩⟨∫_{(2)}, (y ⇀ δ)⟩ = ⟨∫ , f(y ⇀ δ)⟩
= ∫ f(x)δ(xy)dx

since ⟨x, (y ⇀ δ)⟩ = ⟨xy, δ⟩.