quantum group Fourier transform

Quantum group Fourier transform refers to several variants of Fourier transforms attached to Hopf algebras or their analytic versions.

The usual Fourier transform is between functions on locally compact abelian group GG and functions on its Pontrjagin dual locally compact abelian group G^\hat{G} (the group of continuous characters on GG with the topology of uniform convergence on compact sets), cf. wikipedia, Pontrjagin duality.

In noncommutative case, the role of Pontrjagin duality is played by a version of Tannaka-Krein duality.

To avoid analytic details for starters, and still to introduce elements of the general story, one can look at the case when GG is finite. Then, over a field kk, the group Hopf algebra kG^k \hat{G} is isomorphic with the dual Hopf algebra (kG) *(k G)^* of the group Hopf algebra kGk G which is itself isomorphic as a Hopf algebra to the function algebra k(G)k(G). The composition is giving the Fourier transform kG^k(G)k \hat{G}\cong k(G), which is a linear map from the convolution algebra of GG to the function algebra on GG. In such a situation one has a discrete version of Haar measure. Define

(h)(u):= χG^h(χ)χ(u) \mathcal{F}(h)(u) := \sum_{\chi\in \hat{G}} h (\chi)\chi(u)
1(ϕ)(χ):=1|G| uGχ(u 1)ϕ(u) \mathcal{F}^{-1}(\phi)(\chi) := \frac{1}{|G|}\sum_{u\in G}\chi(u^{-1})\phi(u)
Λ= uGukG \Lambda = \sum_{u\in G} u \in k G
Λ *= χχ \Lambda^* = \sum_{\chi} \chi

where one assumes that |G||G| is invertible in kk. It holds that

χΛ *=ϵ(χ)Λ *\chi \Lambda^* = \epsilon(\chi)\Lambda^* for all χkG^\chi\in k \hat{G} and Λϕ=Λϵ(ϕ)\Lambda \phi = \Lambda \epsilon(\phi) for all ϕkG\phi\in k G. In other words, Λ\Lambda is a right integral in kGk G and Λ *\Lambda^* is a left integral in (kG) *(k G)^*.

One can write

(h)=Λ (1)h,Λ (2), \mathcal{F}(h) = \Lambda_{(1)}\langle h, \Lambda_{(2)}\rangle,

for hkG^=(kG) *h\in k \hat{G} = (k G)^* (notice the usage of left coregular action) and

1(ϕ)=Λ (1) *Λ (2) *,SϕΛ *,Λ \mathcal{F}^{-1}(\phi) = \frac{\Lambda^*_{(1)}\langle \Lambda^*_{(2)},S \phi\rangle}{\langle \Lambda^*, \Lambda \rangle}

for ϕkG\phi\in k G and where S:kG(kG) op,copS: k G\to (k G)^{op,cop} is the antipode. These formulas make sense for more general Hopf algebras in duality provided there are appropriate analogues of Λ\Lambda and Λ *\Lambda^* and Λ *,Λ\langle \Lambda^*, \Lambda\rangle is invertible in kk. That generalization is called the quantum group Fourier transform.

They can also be related to the fundamental operator in Hopf algebra HH, see under multiplicative unitary.


For an abelian tensor category we investigate a Hopf algebra F in it, the “algebra of functions” or “automorphisms of the identity functor”. We show the existence of the object of integrals for any Hopf algebra in a rigid abelian category. If some assumptions of finiteness and non-degeneracy are satisfied, the Hopf algebra F has an integral and there are morphisms S,T:FFS, T : F \to F, called modular transformations. They yield a representation of a modular group. The properties of SS are similar to those of the Fourier transform.