## Lecture Notes

### Lie Groups, Lie Algebras, and Representation Theory

#### Lecture Notes

#### Lie Groups and Lie Algebras

#### Representation Theory

#### Problem Sets

#### Lie Groups and Lie Algebras

#### Representation Theory

#### References and Further Reading

Click on linked topics to view lecture notes.

Matrix Lie Groups

The Lie Algebra of the Orthogonal Group \(\mathrm{O}(n)\) (\(\mathrm{SO}(n)\))

Orthogonal Group \(\mathrm{O}(n)\) and Symmetry

The Lorentz Group

Lie Brackets (for \(n\times n\) Matrices)

More Examples of Lie Groups: 3-Sphere as a Lie Group, The 3-Dimensional Heisenberg Group

Lie Group and Lie Algebra Representations

Quantum Angular Momentum and \(\mathfrak{su}(2)\) Representation

Quantum Angular Momentum in \(\mathbb{R}^{2+2}\) and \(\mathfrak{su}(1,1)\)

Lie Group Actions and Lie Group Representations

Irreducible Representations of \(\mathrm{U}(1)\)

The Representation of \(\mathrm{SU}(2)\)

Click on the following links for problems.

Michael F. Atiyah, R. Bott, S. Helgason, D. Kazhdan, B. Kostant, G. Lusztig, I.G. Macdonald, G.W. Mackey, W. Schmid and D.J. Simms, Representation Theory of Lie Groups, Proceedings of the SRC/LMS Research Symposium on Representations of Lie Groups, Oxford, 28 June-15 July 1977, Cambridge University Press, 1979

John Baez and Javier P. Muniain, Gauge Fields, Knots and Gravity, World Scientific, 1994

Andrew Baker, Matrix Groups, An Introduction to Lie Group Theory, Springer_verlag, 2001

Robert L. Bryant, An Introduction to Lie Groups and Symplectic Geometry, A series of nine lectures on Lie groups and symplectic geometry delivered at the Regional Geometry Institute in Park City, Utah, 24 Junes-20 July 1991

Robert N. Cahn, Semi-Simple Lie Algebras and Their Representations, Benjamin-Cummings, 1984

Brian C. Hall, An Elementary Introduction to Groups and Representations

Brian C. Hall, Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, v. 222, Corr. 2nd Printing, Springer, 2004

James E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer-Verlag, 1972

H. F. Jones, Groups, Representations and Physics, Institute of Physics Publishing, 1998

Dong Hoon Lee, The structure of complex Lie groups, Chapman & Hall/CRC, 2002

Barry Simon, Representations of Finite and Compact Groups, American Mathematical Society, 1996

Shlomo Sternberg, Group Theory and Physics, Cambridge University Press, 1994

Peter Woit, Quantum Theory, Groups and Representations: An Introduction