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 $G$ and functions on its Pontrjagin dual locally compact abelian group $\hat{G}$ (the group of continuous characters on $G$ 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 $G$ is finite. Then, over a field $k$, the group Hopf algebra $k \hat{G}$ is isomorphic with the dual Hopf algebra $(k G)^*$ of the group Hopf algebra $k G$ which is itself isomorphic as a Hopf algebra to the function algebra $k(G)$. The composition is giving the Fourier transform $k \hat{G}\cong k(G)$, which is a linear map from the convolution algebra of $G$ to the function algebra on $G$. In such a situation one has a discrete version of Haar measure. Define

$\mathcal{F}(h)(u) := \sum_{\chi\in \hat{G}} h (\chi)\chi(u)$

$\mathcal{F}^{-1}(\phi)(\chi) := \frac{1}{|G|}\sum_{u\in G}\chi(u^{-1})\phi(u)$

$\Lambda = \sum_{u\in G} u \in k G$

$\Lambda^* = \sum_{\chi} \chi$

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

$\chi \Lambda^* = \epsilon(\chi)\Lambda^*$ for all $\chi\in k \hat{G}$ and $\Lambda \phi = \Lambda \epsilon(\phi)$ for all $\phi\in k G$. In other words, $\Lambda$ is a right integral *in* $k G$ and $\Lambda^*$ is a left integral in $(k G)^*$.

One can write

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

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

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

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

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

- V. Lyubashenko,
*Modular transformations and tensor categories*, J. Pure Appl. Algebra**98**(1995) 279–327 doi

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 $S, T : F \to F$, called modular transformations. They yield a representation of a modular group. The properties of $S$ are similar to those of the Fourier transform.

- V. Lyubashenko, S. Majid,
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*Fourier transform and the Verlinde formula for the quantum double of a finite group*, J. Phys. A32: 48 doi - Shahn Majid,
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