Quantum Theory, Groups and Representations: An Introduction
by Peter Woit · Columbia University / Springer
The best entry rung for this topic: the only serious text that motivates unitary representations from the quantum-mechanics angle the topic note names, and fully free. Woit's Columbia course notes, later published by Springer, develop quantum mechanics from unitary representations of Lie groups. Covers the Heisenberg group, Stone-von Neumann theorem, spin, the metaplectic representation, and free-particle representations of the Poincare group.
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Wolfram MathWorld
MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading.
A Course in Abstract Harmonic Analysis (Second Edition)
The standard reference for unitary representations of locally compact groups. Covers Banach and C*-algebras, Haar measure, the Gelfand transform, induced representations and Mackey's imprimitivity theorem, the Plancherel theorem, and representations of semisimple Lie groups. This is the book practitioners cite for the unitary dual and Mackey's induced-representation machinery, and nothing free covers that ground at comparable depth.
Principles of Harmonic Analysis (Second Edition)
Graduate text developing harmonic analysis on locally compact groups: Haar measure, Pontryagin duality, the Plancherel theorem, and the Peter-Weyl theorem, then unitary representations of non-abelian groups, direct integrals, and applications to Heisenberg manifolds and Riemann surfaces.
Lie Groups and Lie Algebras II (MIT 18.755)
Second semester of the graduate Lie theory sequence: Haar measure, representations of compact groups, the Peter-Weyl theorem, maximal tori, and real reductive groups. Includes an open textbook of lecture notes plus problem sets; readers finish able to work with unitary representations of compact Lie groups.
Compact Lie Groups (Graduate Texts in Mathematics 235)
The cleanest standalone treatment of the Peter-Weyl theorem. A 198-page graduate text on compact Lie groups that blends algebra, analysis, and topology. Builds Schur orthogonality, the Peter-Weyl theorem, the Plancherel theorem, maximal tori, Weyl character formulas, highest weight classification, and the Borel-Weil theorem for matrix groups.
nLab
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