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 and functions on its Pontrjagin dual locally compact abelian group (the group of continuous characters on 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 is finite. Then, over a field , the group Hopf algebra is isomorphic with the dual Hopf algebra of the group Hopf algebra which is itself isomorphic as a Hopf algebra to the function algebra . The composition is giving the Fourier transform , which is a linear map from the convolution algebra of to the function algebra on . In such a situation one has a discrete version of Haar measure. Define
where one assumes that is invertible in . It holds that
for all and for all . In other words, is a right integral in and is a left integral in .
One can write
for (notice the usage of left coregular action) and
for and where is the antipode. These formulas make sense for more general Hopf algebras in duality provided there are appropriate analogues of and and is invertible in . That generalization is called the quantum group Fourier transform.
They can also be related to the fundamental operator in Hopf algebra , which is the invertible operator satisfying the pentagon identity
in the tensor cube of the space of linear endomorphisms of and such that
for all . For finite-dimensional Hopf algebras and .
- 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 quantum gravity, Classical Quantum Gravity 25 (2008), no. 4, 045006 MR2009f:83058, doi