Matrix Lie Groups Course | Introduction, Syllabus & Applications | IISER Mohali
Course Details
| Exam Registration | 18 |
|---|---|
| Course Status | Ongoing |
| Course Type | Elective |
| Language | English |
| Duration | 12 weeks |
| Categories | Mathematics |
| Credit Points | 3 |
| Level | Undergraduate/Postgraduate |
| Start Date | 19 Jan 2026 |
| End Date | 10 Apr 2026 |
| Enrollment Ends | 02 Feb 2026 |
| Exam Registration Ends | 20 Feb 2026 |
| Exam Date | 19 Apr 2026 IST |
| NCrF Level | 4.5 — 8.0 |
An Introduction to Matrix Lie Groups: A Foundational Course
Delve into the elegant world where algebra meets geometry. This detailed introduction to a 12-week course on Matrix Lie Groups, offered by the distinguished Prof. Krishnendu Gongopadhyay of IISER Mohali, provides a roadmap for students to master this fundamental area of modern mathematics.
About the Instructor: Prof. Krishnendu Gongopadhyay
Prof. Krishnendu Gongopadhyay is a leading mathematician and professor at the Indian Institute of Science Education and Research (IISER) Mohali, specializing in geometry and group theory. A Fellow of the National Academy of Sciences, India (NASI, 2023), his academic journey includes a Ph.D. from IIT Bombay and postdoctoral work at premier institutes like TIFR and ISI Kolkata. With over a decade at IISER Mohali, he has guided numerous Ph.D. and M.S. students, and is deeply committed to mathematical outreach and education. His expertise ensures this course is both rigorous and accessible.
Course Overview
This course serves as a gateway to the theory of Matrix Lie Groups—groups of invertible matrices that act as transformations of space. It brilliantly explores the interplay between group theory, geometry, and linear algebra, building essential tools for advanced study in differential geometry, Lie theory, and mathematical physics. Designed with clarity in mind, the course requires only foundational knowledge, making it perfect for motivated undergraduates and postgraduates.
Who Should Take This Course?
Level: Undergraduate/Postgraduate
Prerequisites: A solid understanding of Calculus, basic Linear Algebra, and elementary Group Theory. Familiarity with general topology is helpful but not mandatory.
Industry Relevance
The concepts taught in this course are not just theoretical. They form the mathematical backbone of several cutting-edge technologies. Industries involved in Quantum Computing, Robotics, and Control Theory will find the skills developed here directly applicable.
Detailed 12-Week Course Layout
| Week | Topics Covered |
|---|---|
| Week 1 | Geometry of complex numbers. The quaternions. Motivating examples: Space rotations and Rotations of a sphere. |
| Week 2 | The general linear groups. Conjugation and Change of basis. All matrix groups are real matrix groups. |
| Week 3 | The Unitary groups. The Euclidean isometry group. Low dimensional examples. |
| Week 4 | Topology of matrix groups. Open sets. Continuity. Connected sets. Compact sets. |
| Week 5 | Lie algebras. Examples. Lie algebras as vector fields. The Lie algebras of orthogonal groups. |
| Week 6 | Matrix Exponentiation. Properties of Matrix Exponentiation. One parameter subgroups. |
| Week 7 | Analysis background. Differentiation. Chain rules. Inverse and Implicit function theorems. |
| Week 8 | Restriction of exponential map to Lie algebras. Realization of matrix groups as smooth manifolds. |
| Week 9 | The Lie bracket. Adjoint representation. Example of Adjoint representation. |
| Week 10 | The double cover Sp(1) → SO(3), Some other double covers. Sketch of the Lie group-Lie algebra correspondence. |
| Week 11 | Maximal torus. Center of compact matrix groups. Conjugates of maximal tori. |
| Week 12 | Introduction to smooth manifolds and Lie groups. Example that not all Lie groups are matrix groups. Applications of Lie groups in Natural sciences. |
Recommended Textbooks
To complement the lectures, the following texts are highly recommended:
- Kristopher Tapp, Matrix Groups for Undergraduates (2nd Ed.) - An excellent primary text for the course's level.
- M. L. Curtis, Matrix Groups - A classic, deeper treatment of the subject.
- Robert Gilmore, Lie Groups, Physics, and Geometry - Perfect for understanding the physical applications and broader context.
Why Study Matrix Lie Groups?
Matrix Lie Groups are the "continuous symmetry" groups of mathematics. They provide the language to describe rotations, Lorentz transformations in relativity, gauge symmetries in particle physics, and motions in robotic systems. This course doesn't just teach definitions; it builds an intuitive geometric understanding, connects local algebra (Lie algebras) to global group structure, and opens doors to vast areas of pure and applied mathematics. Under the guidance of an expert like Prof. Gongopadhyay, students will gain not just knowledge, but a profound appreciation for the beauty and utility of this field.
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