Physics Inspired Introduction to Lie Groups and Lie Algebras

The study of Lie groups and Lie algebras—mathematical structures that capture notions of symmetry—and their representations, which express these symmetries concretely, has been deeply influenced by developments in physics. One one hand, the theory of compact Lie groups and semisimple Lie algebras was significantly motivated by quantum mechanics. On the other hand, the theory of non-compact Lie groups was driven by the demands of relativity and relativistic quantum mechanics.

This course introduces representation theory of Lie groups from first principles, clearly illustrating how it naturally emerges from questions in quantum mechanics and relativity. Subsequently, the course develops a semi-rigorous mathematical framework, exploring typical problems and some of their solutions within this rich interplay of physics and mathematics.

Instructor 1 : Satwata Hans (Webpage)

Doctoral student, Penn State University, USA

Instructor 2 : Sayak Biswas (Webpage)

Doctoral student, Ohio State University, USA

Pre-requisites

Basic notions of Quantum mechanics (on the level of second year IISER Kolkata class).
Linear algebra (on the level of the second year IISER Kolkata class).
Basic notions of groups (on the level of the second year IISER Kolkata class).

Schedule

July 1, 2025 to July 24, 2025.
Tuesdays and Thursdays - 10:00 am to 11:30 am.
Classes to be held over Zoom. Please email either Satwata or Sayak for the zoom link.

Plan (All lectures are planned to last 1.5 hours.)

Lecture 1 (Satwata Hans): Introduction to representation theory of finite groups through the symmetry group of the triangle. (Lecture notes)
Lecture 2 (Satwata Hans): Introduction to abstract harmonic analysis on finite groups. The Fourier transform on finite groups, Plancherel's formula and the Peter-Weyl theorem for finite groups. (Lecture notes)
Lecture 3 (Sayak Biswas): Why do we care about representations - a Quantum physicist's perspective through angular momentum, the rotation group SO(3) and the special unitary group SU(2).
Lectures 4 (Satwata Hans): Basic notions of Lie groups and Lie algebras and their representations. How does SU(2) differ from SO(3), and what does it mean for their representations?
Lecture 5 (Satwata Hans): Finding a complete list of representations of certain complex Lie algebras.
Lecture 6 (Satwata Hans): A deep dive into representations of compact Lie groups. Peter-Weyl theorem and the Plancherel theorem for compact Lie groups.
Lecture 7 (Sayak Biswas): Reasons to think beyond finite dimensional irreducible representations: Lorentz group and relativistic quantum mechanics.
Lectures 8 (Satwata Hans): A brief tour into the representation theory of non-compact groups. Harish Chandra's Plancherel theorem.

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