Quantization maps, algebra representation and non-commutative Fourier transform for Lie groups

TL;DR

This paper explores the algebraic and analytical structures underlying non-commutative Fourier transforms on Lie groups, linking quantization maps to star-products and providing explicit examples for U(1) and SU(2).

Contribution

It establishes a framework connecting quantization maps with algebra representations and non-commutative Fourier transforms on Lie groups, including explicit constructions for specific cases.

Findings

Derived conditions for algebra representation existence from quantization maps.

Explicit star-products and non-commutative plane waves for U(1) and SU(2).

Clarified the link between quantization, algebra representation, and Fourier analysis.

Abstract

The phase space given by the cotangent bundle of a Lie group appears in the context of several models for physical systems. A representation for the quantum system in terms of non-commutative functions on the (dual) Lie algebra, and a generalized notion of (non-commutative) Fourier transform, different from standard harmonic analysis, has been recently developed, and found several applications, especially in the quantum gravity literature. We show that this algebra representation can be defined on the sole basis of a quantization map of the classical Poisson algebra, and identify the conditions for its existence. In particular, the corresponding non-commutative star-product carried by this representation is obtained directly from the quantization map via deformation quantization. We then clarify under which conditions a unitary intertwiner between such algebra representation and the usual group representation can be constructed giving rise to the non-commutative plane waves and consequently, the non-commutative Fourier transform. The compact groups U(1) and SU(2) are considered for different choices of quantization maps, such as the symmetric and the Duflo map, and we exhibit the corresponding star-products, algebra representations and non-commutative plane waves.