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\begin{document}

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\section*{quantum group Fourier transform}

\textbf{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, \href{http://en.wikipedia.org/wiki/Pontryagin_duality}{Pontrjagin duality}.

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

\begin{displaymath}
\mathcal{F}(h)(u) := \sum_{\chi\in \hat{G}} h (\chi)\chi(u)
\end{displaymath}
\begin{displaymath}
\mathcal{F}^{-1}(\phi)(\chi) := \frac{1}{|G|}\sum_{u\in G}\chi(u^{-1})\phi(u)
\end{displaymath}
\begin{displaymath}
\Lambda = \sum_{u\in G} u \in k G
\end{displaymath}
\begin{displaymath}
\Lambda^* = \sum_{\chi} \chi
\end{displaymath}
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 \emph{in} $k G$ and $\Lambda^*$ is a left integral in $(k G)^*$.

One can write

\begin{displaymath}
\mathcal{F}(h) = \Lambda_{(1)}\langle h, \Lambda_{(2)}\rangle,
\end{displaymath}
for $h\in k \hat{G} = (k G)^*$ (notice the usage of left [[coregular action]]) and

\begin{displaymath}
\mathcal{F}^{-1}(\phi) = \frac{\Lambda^*_{(1)}\langle \Lambda^*_{(2)},S \phi\rangle}{\langle \Lambda^*, \Lambda \rangle}
\end{displaymath}
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 [[dual bialgebra|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 \textbf{quantum group Fourier transform}.

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

\hypertarget{literature}{}\subsubsection*{{Literature}}\label{literature}

\begin{itemize}%
\item V. Lyubashenko, \emph{Modular transformations and tensor categories}, J. Pure Appl. Algebra \textbf{98} (1995) 279–327 

\end{itemize}
\begin{quote}%
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.


\end{quote}
\begin{itemize}%
\item V. Lyubashenko, S. Majid, \emph{Fourier transform identities in quantum mechanics and the quantum line}, Phys. Lett. B284 (1992) 66-70 ; \emph{Braided groups and quantum Fourier transform}, J. Algebra 166 506-28 \href{https://doi.org/10.1006/jabr.1994.1165}{doi}
\item T H Koornwinder, B J Schroers, J K Slingerland, F A Bais, \emph{Fourier transform and the Verlinde formula for the quantum double of a finite group}, J. Phys. A32: 48 \href{https://doi.org/10.1088/0305-4470/32/48/313}{doi}
\item [[Shahn Majid]], \emph{Foundations of quantum group theory}, 1995, 2nd. ed 2000
\item M. Enock, J. M. Schwartz, \emph{Kac algebras and duality of locally compact groups}, Springer-Verlag, 1992, , x+257 pp. \href{http://books.google.com/books/about/Kac_algebras_and_duality_of_locally_comp.html?id=U6e6aD1gj3oC}{gBooks}, \href{http://www.ams.org/mathscinet-getitem?mr=1215933}{MR94e:46001}
\item Massoud Amini, \emph{Tannaka--Krein duality for compact groupoids I, Representation theory}, Advances in Mathematics \textbf{214}, n. 1, 2007, 78-91 \href{http://dx.doi.org/10.1016/j.aim.2006.09.015}{doi}
\item Laurent Freidel, [[nlab:Shahn Majid]], \emph{Noncommutative harmonic analysis, sampling theory and the Duflo map in $2+1$ quantum gravity}, Classical Quantum Gravity \textbf{25} (2008), no. 4, 045006 \href{http://www.ams.org/mathscinet-getitem?mr=2388191}{MR2009f:83058}, \href{http://dx.doi.org/10.1088/0264-9381/25/4/045006}{doi}
\item [[zoranskoda:ncFourier]]

\end{itemize}


\end{document}