8.1 The Peter-Weyl theorem and harmonic analysis

The Peter-Weyl theorem connects representation theory and harmonic analysis for compact Lie groups. It shows that matrix coefficients of irreducible unitary representations form a complete orthonormal basis for square-integrable functions on the group.

This powerful result allows us to decompose functions on compact Lie groups into sums of matrix coefficients. It generalizes Fourier analysis to groups like SU(2) and SO(3), linking their algebraic structure to properties of functions on the group.

Peter-Weyl Theorem in Harmonic Analysis

Fundamental Result in Harmonic Analysis on Compact Lie Groups

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• Establishes a connection between the representation theory of a compact Lie group and the space of square-integrable functions on the group
• States that the matrix coefficients of irreducible unitary representations of a compact Lie group form a complete orthonormal basis for the space of square-integrable functions on the group
• Allows for the decomposition of functions on a compact Lie group into a sum of matrix coefficients, the building blocks of the group's representation theory
• Generalizes the classical Fourier analysis on the circle group to arbitrary compact Lie groups (SU(2), SO(3))

Studying Compact Lie Groups through Representation Theory and Harmonic Analysis

• Provides a way to study the structure of compact Lie groups through their representation theory and harmonic analysis
• Enables the application of techniques from representation theory to problems in harmonic analysis, such as the study of convolution operators and the Fourier transform on compact Lie groups
• Offers a unified framework for understanding the interplay between the algebraic structure of a compact Lie group (through its representations) and the analytic properties of functions on the group (through harmonic analysis)
• Can be applied to specific compact Lie groups (special unitary group SU(n), special orthogonal group SO(n)) to study their representation theory and harmonic analysis in detail

Decomposing Functions with Peter-Weyl Theorem

Decomposing Square-Integrable Functions into Irreducible Representations

• Allows for the decomposition of a square-integrable function on a compact Lie group into a sum of matrix coefficients of irreducible unitary representations
• Given a square-integrable function f on a compact Lie group G, expresses f as a sum of matrix coefficients:
  • $f(g) = \Sigma_\pi d_\pi \Sigma_{i,j} <f, \pi_{ij}> \pi_{ij}(g)$, where $\pi$ runs over all irreducible unitary representations of G, $d_\pi$ is the dimension of the representation $\pi$, and $\pi_{ij}$ are the matrix coefficients of $\pi$
• Coefficients $<f, \pi_{ij}>$ in the decomposition are given by the inner product of the function f with the matrix coefficients $\pi_{ij}$ in the space of square-integrable functions on G
• Allows for the study of the Fourier analysis on compact Lie groups, where the irreducible representations play the role of the Fourier basis (exponential functions, spherical harmonics)

Analyzing Function Properties through Representation Theory

• Decomposition of a function into irreducible representations provides a way to analyze the function's properties and behavior in terms of the group's representation theory
• Enables the study of functions on a compact Lie group through the lens of representation theory
• Allows for the analysis of symmetries and invariance properties of functions on the group, which reflect the underlying structure of the group
• Provides insights into the structure and properties of the group itself by studying the dimensions and matrix coefficients of the irreducible representations

Peter-Weyl Theorem: Representation Theory and Harmonic Analysis

Bridging Representation Theory and Harmonic Analysis

• Bridges the gap between representation theory and harmonic analysis on compact Lie groups
• Shows that the irreducible unitary representations of a compact Lie group form a complete orthonormal basis for the space of square-integrable functions on the group
• Connects the algebraic structure of a compact Lie group (through its representations) with the analytic properties of functions on the group (through harmonic analysis)
• Enables the application of representation theory techniques to harmonic analysis problems (convolution operators, Fourier transform)

Constructing Fourier Transform on Compact Lie Groups

• Allows for the construction of a Fourier transform on compact Lie groups
• Fourier transform on compact Lie groups can be used to study the group's structure through its Fourier coefficients
• Decomposition of functions into irreducible representations provided by the Peter-Weyl theorem serves as the basis for the Fourier transform
• Fourier transform on specific compact Lie groups (SU(2), SO(3)) can reveal important properties and symmetries of the group

Studying Compact Lie Groups with Peter-Weyl Theorem

Analyzing Irreducible Unitary Representations

• Can be used to study the structure of compact Lie groups by analyzing their irreducible unitary representations
• Implies that the irreducible unitary representations of a compact Lie group G completely determine the structure of the space of square-integrable functions on G
• Studying the dimensions and matrix coefficients of the irreducible representations provides insights into the structure and properties of the group itself
• Allows for the analysis of symmetries and invariance properties of functions on the group, reflecting the underlying group structure

Applying Peter-Weyl Theorem to Specific Compact Lie Groups

• Can be applied to specific compact Lie groups to study their representation theory and harmonic analysis in detail
• Examples of compact Lie groups:
  • Special unitary group SU(n): group of n x n unitary matrices with determinant 1
  • Special orthogonal group SO(n): group of n x n orthogonal matrices with determinant 1
• Studying the irreducible representations and Fourier analysis of these groups reveals their unique properties and symmetries
• Applying the Peter-Weyl theorem to SU(2) and SO(3) provides insights into the representation theory and harmonic analysis of rotations in three-dimensional space