PeterWeyl

Peter Weyl, often encountered in the form Peter–Weyl, was a German mathematician whose work helped establish modern representation theory and harmonic analysis on compact groups. He is best known for results that culminate in what is now called the Peter–Weyl theorem, a foundational result in the representation theory of compact groups.

The Peter–Weyl theorem provides a concrete Fourier-type framework for compact groups. In the setting of a compact group G, the matrix coefficients of every finite-dimensional unitary representation separate points and form a dense subspace of the space C(G) of continuous functions on G. Equivalently, the regular representation of G on L2(G) decomposes into a direct sum of finite-dimensional irreducible representations. These conclusions yield a powerful tool for analyzing functions on G and for understanding the representations of the group.

Beyond this theorem, Weyl contributed to the broader theory of Lie groups and harmonic analysis. He is associated with ideas that led to the Weyl character formula, which provides explicit expressions for characters of irreducible representations of compact Lie groups, and with foundational developments in the structure and representations of Lie groups.

Biographical details of Weyl’s life are less prominent in popular summaries, but his work is embedded in the early 20th-century development of representation theory and its connections to analysis and geometry. His influence persists in many areas of mathematics and mathematical physics.

The Peter–Weyl theorem remains a standard result in modern mathematics, shaping approaches in representation theory, harmonic analysis, operator algebras, and quantum theory, and it is commonly presented in graduate courses and reference texts on the subject.