# Fourier transform, Schr\"odinger representation, and Heisenberg modules

## TL;DR

This paper explores the role of Fourier transform and Schr"odinger representation in analyzing twisted group algebras, revealing relations with Heisenberg modules and applications to noncommutative solitons.

## Contribution

It introduces a dual equivalence bimodule linking Schr"odinger representations and Fourier transform within the context of twisted group algebras.

## Key findings

- Construction of dual equivalence bimodule for Heisenberg bimodule
- Relations between Schr"odinger representation and Fourier transform
- Application of noncommutative solitons

## Abstract

We investigate and review how Fourier transform is involved in the analysis of a twisted group algebra $L^1(G, \sigma)$ for $G=\widehat{\Gamma}\times \Gamma$ and $\sigma:G\times G \to \mathbb{T}$ 2- cocycle where $\Gamma$ is a locally compact abelian group and $\widehat{\Gamma}$ its Pontryagin dual. By weaving the Schr\"{o}dinger representation and Fourier transform, we construct the dual equivalence bimodule of the Heisenberg bimodule generated by the dual Schr\"{o}dinger representation and observe several relations between them including the application of noncommutative solitons.