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 $H$, which is the invertible operator $W : H\otimes H \to H\otimes H$ satisfying the pentagon identity

$W_{1 2} W_{1 3} W_{2 3} = W_{2 3} W_{1 2}$

in the tensor cube of the space of linear endomorphisms of $H$ and such that

$\Delta(h) = W (h\otimes 1) W^{-1}$

$S h = (\epsilon\otimes id)\circ W^{-1}(h\otimes - )$

for all $h\in H$. For finite-dimensional Hopf algebras $W(g\otimes h) = g_{(1)}\otimes g_{(2)} h$ and $W^{-1}(g\otimes h = g_{(1)}\otimes (S g_{(2)}) h$.

- Shahn Majid,
*Foundations of quantum group theory*, 1995, 2nd. ed 2000 - M. Enock, J. M. Schwartz,
*Kac algebras and duality of locally compact groups*, Springer-Verlag, 1992, , x+257 pp. gBooks, MR94e:46001 - Massoud Amini,
*Tannaka–Krein duality for compact groupoids I, Representation theory*, Advances in Mathematics**214**, n. 1, 2007, 78-91 doi - Laurent Freidel, Shahn Majid,
*Noncommutative harmonic analysis, sampling theory and the Duflo map in $2+1$ quantum gravity*, Classical Quantum Gravity**25**(2008), no. 4, 045006 MR2009f:83058, doi - ncFourier

Revised on May 16, 2013 20:01:04
by Zoran Škoda
(161.53.130.104)