nLab
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 $(kG{)}^{*}$ of the group Hopf algebra $kG$ 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

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

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

$$\Lambda =\sum _{u\in G}u\in kG$$\Lambda = \sum_{u\in G} u \in k G

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

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

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

One can write

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

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

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

for $\varphi \in kG$ and where $S:kG\to (kG{)}^{\mathrm{op},\mathrm{cop}}$ is the antipode. These formulas make sense for more general Hopf algebras in duality provided there are appropriate analogues of $\Lambda$ and ${\Lambda }^{*}$ and $⟨{\Lambda }^{*},\Lambda ⟩$ 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

$$W_{12}{W}_{13}{W}_{23}=W_{23}{W}_{12}$$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

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

$$Sh=(ϵ\otimes \mathrm{id})\circ W^{-1}(h\otimes -)$$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