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

W 12W 13W 23=W 23W 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 HH and such that

Δ(h)=W(h1)W 1 \Delta(h) = W (h\otimes 1) W^{-1}
Sh=(ϵid)W 1(h) S h = (\epsilon\otimes id)\circ W^{-1}(h\otimes - )

for all hHh\in H. For finite-dimensional Hopf algebras W(gh)=g (1)g (2)hW(g\otimes h) = g_{(1)}\otimes g_{(2)} h and W 1(gh=g (1)(Sg (2))hW^{-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+12+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)