The Fourier transform in quantum group theory

Abstract

The Fourier transform, known in classical analysis, and generalized in abstract harmonic analysis, can also be considered in the theory of locally compact quantum groups. In this note, I discuss some aspects of this more general Fourier transform. In order to avoid technical difficulties, typical for the analytical approach, I will restrict to the algebraic quantum groups. Roughly speaking, these are the locally compact quantum groups that can be treated with purely algebraic methods (in the framework of multiplier Hopf algebras). I will illustrate various notions and results using not only classical Fourier theory on the circle group $\mathbb{T}$, but also on the additive group ${\mathbb{Q}}_p$ of $p$-adic numbers. It should be observed however that these cases are still too simple to illustrate the full power of the more general theory.

References in corpus (2)

• Compact and discrete subgroups of algebraic quantum groups I
• Multiplier Hopf algebras imbedded in C${}^{\ast }$-algebraic quantum groups

Cited by in corpus (9)

• On idempotent states on quantum groups
• Symmetries of Lévy processes on compact quantum groups, their Markov semigroups and potential theory
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• Fourier transform on locally compact quantum groups
• Uncertainty Principles for Kac Algebras
• Twisting of the Quantum double and the Weyl algebra
• Spherical Fourier Transforms on Locally Compact Quantum Gelfand Pairs
• Abstract Harmonic Analysis on Spacetime
• Plancherel Theorem and the Left Ideals of the Group Algebra for the Jacobi Group