FOURIER TRANSFORMS OVER FINITE QUANTUM GROUPS

BODO PAREIGIS

Herrn Professor Dr. D. Pumplün als Dank für sein langjähriges Engagement für die Mathematik und die Universität gewidmet.

1. Introduction

In this note we want to clarify the notion of an integral for arbitrary Hopf algebras that has been introduced a long time ago [2, 6]. The relation between the integral on a Hopf algebra and integrals in functional analysis has only been hinted at in several publications. With the strong interest in quantum groups, i.e. non-commutative and non-cocommutative Hopf algebras, we wish to show in which form certain transformation rules for integrals occur in quantum groups.

Our point of view will be the following. Let G be a quantum group in the sense of non-commutative algebraic geometry, that is a space whose function algebra is given by an arbitrary Hopf algebra H over some base field 𝕂. We will also have to use the algebra of linear functionals H^{∗} = Hom(H, 𝕂) with the multiplication induced by the diagonal of H (called the bialgebra of G in the French literature). For most of this paper we will assume that H is finite dimensional. Observe that the functions in H do not commute under multiplication and that they usually have no general commutation formula.

The model for this setup can be found in functional analysis. There the group G is a locally compact group, H the space of representative functions on G, and H^{∗} the space of generalized functions or distributions. Then the functions commute under multiplication.

We will also consider two special examples of our setup. For an arbitrary finite group G the Hopf algebra H = 𝕂^{G} is defined to be the algebra of functions on G. Then H^{∗} = 𝕂G, the group algebra, is the linear dual of H.

If the finite group G is Abelian and if 𝕂 is algebraicly closed with char(𝕂) −̸ |G| then the corresponding Hopf algebra is as above H = 𝕂^{G} and H^{∗} = 𝕂G. By Pontryagin duality there is the group Ĝ of characters on G such that H = 𝕂^{G} = 𝕂Ĝ and H^{∗} = 𝕂G = 𝕂Ĝ.

I would like to acknowledge stimulating conversations on the subject of this note with H. Röhrl and Y. Sommerhäuser.

Date: February 10, 1998.

1991 Mathematics Subject Classification. Primary 16A10.

2. Integrals

Let H be an arbitrary Hopf algebra [3, 6]. The linear functionals a ∈ H^{∗} will be considered as generalized integrals on H ([5] p.123). We have an operation H^{∗} ⊗ H ∋ a ⊗ f ↦ ⟨a, f⟩ ∈ 𝕂 that 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.

We will be interested in a special generalized integral ∫ ∈ H^{∗} satisfying

a∫ = ⟨a, 1_{H}⟩∫ (1)

or a ∫ = ε(a) ∫. Such an integral is called a left invariant integral.

In the case of a locally compact group G such an element is given by the Haar integral with respect to a left invariant Haar measure [1]

∫_{G} f(x)μdx = ⟨∫ , f⟩.

Therefore we write in the general quantum group situation

∫ f(x)dx := ⟨∫ , f⟩. (2)

This notation has two parentheses, ∫ and dx, so that the integrand f is clearly separated. We also use the notation

∫ f(x)g(x)dx := ⟨∫ , fg⟩. (3)

Observe that f(x) and g(x) are just parts of the whole symbol and in particular that they do not commute.

In the case of a finite group G a left invariant integral in H^{∗} = 𝕂G on H = 𝕂^{G} is known to be

∫ = ∑_{x∈G} x (4)

since y ∑_{x∈G} x = ∑_{x∈G} yx = ∑_{x∈G} x = ⟨y, 1_{H}⟩ ∑_{x∈G} x. For arbitrary a ∈ 𝕂G we have a ∑_{x∈G} x = ε(a) ∑_{x∈G} x. So our integral notation turns out to be

∫ f(x)dx = ∑_{x∈G} f(x) (5)

and has the property

∫ f(x)dx = ∑_{x∈G} f(x) = ∑_{x∈G} f(yx) = ∫ f(yx)dx (6)

for all y ∈ G. This left invariant integral turns out to be also right invariant ∫ a = ∫ ε(a).

3. v. Neumann Transforms

We return to the arbitrary Hopf algebra H of a quantum group. 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} (7)

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⟩. (8)

Analogously we have

(a ↼ f) = ∑⟨a_{(1)}, f⟩a_{(2)}. (9)

An easy observation about left invariant integrals on H is

Lemma 1. The set Int_{l}(H^{∗}) of left invariant integrals is a two sided ideal in H^{∗}.

The integral ∫ is left invariant iff ∀y ∈ H^{∗} : y ∫ = ε(y) ∫ iff ∀y ∈ H^{∗}, f ∈ H : ⟨y ∫ , f⟩ = ⟨∫ , (f ↼ y)⟩ = ε(y)⟨∫ , f⟩. Since ⟨x, f⟩ = f(x) and ⟨x, (f ↼ y)⟩ = ⟨yx, f⟩ = f(yx), the integral ∫ is left invariant iff

∫ f(yx)dx = ε(y) ∫ f(x)dx. (10)

Theorem 2. If there exists 0 ≠ ∫ ∈ Int_{l}(H^{∗}) then the map H ∋ f ↦ (∫ ↽ f) ∈ H^{∗} is injective.

Proof. By [6] theorem 5.1.3 the following homomorphism Int_{l}(H^{∗}) ⊗ H ∋ ∫ ⊗f ↦ (∫ ↽ f) ∈ H^{∗}^{rat}(⊆ H^{∗}) is bijective.

Corollary 3. If there exists 0 ≠ ∫ ∈ Int_{l}(H^{∗}) then the antipode S : H → H is injective.

Proof. The monomorphism H ∋ f ↦ (∫ ↽ f) ∈ H^{∗} is composed of S : H → H and H ∋ f ↦ (f ⇀ ∫) ∈ H^{∗}.

We call a generalized integral a ∈ H^{∗} a rational integral if a is of the form a = ∑ ∫_{i} ↽ f_{i}.

Corollary 4. For every rational integral a ∈ H^{∗} there is a unique g ∈ H such that

⟨a, f⟩ = ∫ f(x)S(g)(x)dx

for all f ∈ H.

Proof. For every rational integral a there is a unique function g ∈ H with a = (∫ ↽ g), hence ⟨a, f⟩ = ⟨∫ ↽ g, f⟩ = ⟨S(g) ⇀ ∫ , f⟩ = ⟨∫ , fS(g)⟩ = ∫ f(x)S(g)(x)dx.

One of the first to study this property of the integral ∫ to represent other linear functionals was J. v. Neumann in [4].

If H is finite dimensional then the isomorphism Int_{l}(H^{∗}) ⊗ H ∋ ∫ ⊗f ↦ (∫ ↽ f) ∈ H^{∗}^{rat}(⊆ H^{∗}) shows Int_{l}(H^{∗}) has dimension 1.

We choose for the rest of this paper a non zero left invariant integral ∫ whenever we are in the situation of H finite dimensional.

Let H be finite dimensional. Since ∫ a is a left invariant integral and dim(Int_{l}(H^{∗})) = 1 there is a unique mod(a) ∈ 𝕂 with

∫ a = mod(a)∫ .

One checks that mod : H^{∗} → 𝕂 is an algebra homomorphism called the modulus of H^{∗}. If mod = ε = 1_{H^{∗}} then H^{∗} is called unimodular. This is equivalent to ∫ also being right invariant or Int_{l}(H^{∗}) = Int_{r}(H^{∗}).

Corollary 5. If H is finite dimensional then for every a ∈ H^{∗} there is a unique g ∈ H such that ⟨a, f⟩ = ∫ f(x)S(g)(x)dx for all f ∈ H.

Corollary 6. If H is finite dimensional then S : H → H and H ∋ f ↦ (f ⇀ ∫) ∈ H^{∗} are bijective.

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

⟨a, f⟩ = ∫ f(x)S(g)(x)dx = ∑_{x∈G} f(x)g(x^{−1}) (11)

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

4. The Nakayama Automorphism

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)⟩ (12)

or

∫ f(x)g(x)dx = ∫ g(x)ν(f)(x)dx. (13)

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 7. 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 8. 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.

5. The Dirac Delta Function

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⟩. (14)

Proposition 9..

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 [6] exercise V.4.

Lemma 10. Let H be a finite dimensional Hopf algebra. Then ∫ ∈ H^{∗} is a left integral iff

a(∑ ∫_{(1)} ⊗ S(∫_{(2)})) = (∑ ∫_{(1)} ⊗ S(∫_{(2)}))a (15) iff

∑ S(a)∫_{(1)} ⊗ ∫_{(2)} = ∑ ∫_{(1)} ⊗ a∫_{(2)} (16)

iff

∑ f_{(1)}⟨∫ , f_{(2)}⟩ = ⟨∫ , f⟩1_{H}. (17)

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 (17) 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} (18)

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 f(e) = 1.

If G is a finite Abelian group we get δ = α ∑_{χ∈Ĝ} χ for some α ∈ 𝕂. The evaluation gives 1 = α⟨∫ , 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}.

FOURIER TRANSFORMS OVER FINITE QUANTUM GROUPS 7

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

δ = |G|^{−1} ∑_{χ∈Ĝ} χ. (19)

6. Fourier Transforms

Let H be finite dimensional for the rest of this paper. In Corollary 8 we have seen that the map H ∋ f ↦ (∫ ↼ f) ∈ H^{∗} is an isomorphism. This map will be called the Fourier transform. Theorem 11. The Fourier transform H ∋ f ↦ f̃ ∈ H^{∗} is bijective with f̃ = (∫ ↼ f) = ∑⟨∫_{(1)}, f⟩∫_{(2)} (20) The inverse Fourier transform is defined by ã = ∑ S^{−1}(δ_{(1)})⟨a, δ_{(2)}⟩. (21) 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} (22)

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⟩ (23) and ⟨f̃, g⟩ = ⟨(∫ ↼ f), g⟩ = ⟨∫ , fg⟩ (24) we get for all a ∈ H^{∗} and f ∈ H

\begin{aligned}⟨a,ˆf̂⟩ & = ⟨af̂, δ⟩ = ∑⟨a, δ_{(1)}⟩⟨f̂, δ_{(2)}⟩ = ∑⟨a, δ_{(1)}⟩⟨∫ , fδ_{(2)}⟩ ( by Lemma 10 ) \\ & = ∑⟨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.

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 (4) and (19) 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} (25)

This implies

f̃(χ) = ∑_{x∈G} f(x)χ(x) = ∫ f(x)χ(x)dx (26)

with inverse transform

ã(x) = |G|^{−1} ∑_{χ∈Ĝ} χ(a)χ^{−1}(x). (27)

Lemma 12. The Fourier transforms of the left invariant integrals in H and H^{∗} are

δ̃ = εν^{−1} ∈ H^{∗}  and  ˜∫ = 1 ∈ H. (28)

Proof. We have ⟨δ̃, f⟩ = ⟨∫ , δf⟩ = ⟨∫ , ν^{−1}(f)δ⟩ = εν^{−1}(f)⟨∫ , δ⟩ = εν^{−1}(f) hence δ̃ = εν^{−1} and ⟨a, ˜∫⟩ = ∑⟨a, S^{−1}(δ_{(1)})⟩⟨∫ , δ_{(2)}⟩ = ⟨a, S^{−1}(1)⟩⟨∫ , δ⟩ = ⟨a, 1⟩, hence ˜∫ = 1.

Proposition 13. Define a convolution multiplication on H^{∗} by

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

Then the following transformation rule holds for f, g ∈ H:

f̃g = f̃∗ g̃. (29)

In particular H^{∗} with the convolution multiplication is an associative algebra with unit ˜1_{H} = ∫, i.e.

∫ ∗ a = a ∗ ∫ = a. (30)

FOURIER TRANSFORMS OVER FINITE QUANTUM GROUPS 9

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 (28) 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}

7. The Plancherel Formula

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 14. (The Plancherel formula) ⟨a, f⟩ = ⟨f̃, ν(ã)⟩. (31) Proof. First we have

⟨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 15. 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 16.

⟨a, f⟩ = ∫ ã(x)f(x)dx = ∫ ^{∗} S^{−1}(a)(x)f̃(x)dx. (32)

Proof. We have the rules for the Fourier transform. From (24) we get ⟨a, f⟩ = ⟨∫ , ãf⟩ = ∫ ã(x)f(x)dx and from (23) we get

⟨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 17. 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⟩.

References

[1] A. Haar: Der Maßbegriff in der Theorie der kontinuierlichen Gruppen. Annals of Math., 34 (147-169), 1933.

[2] R. G. Larson, M. E. Sweedler: An Associative Orthogonal Bilinear Form for Hopf Algebras. Am. J. Math. 91 (75-93), 1969.

[3] S. Montgomery: Hopf Algebras and Their Actions on Rings. CBMS 82, AMS 1993.

[4] J. v. Neumann: On rings of operators. III. Annals of Math., 41 (94-161), 1940.

[5] F. Riesz, B. Sz.-Nagy: Vorlesungen über Funktionalanalysis. VEB Deutscher Verlag der Wissenschaften, Berlin, 1956.

[6] M. E. Sweedler: Hopf Algebras. Benjamin, New York, 1969.

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