Fall 2025
Vladimir Retakh
Course Description:
This course is an introduction to the theory of Lie algebras. Lie algebras are some of the fundamental concepts of algebra, which have proved to be useful in many areas of mathematics and physics. Topics will include elementary properties of Lie algebras, theorems of Lie and Cartan, conjugacy of Borel subalgebras, root systems and their construction, structure and classification of semisimple Lie algebras, and representation theory of semisimple Lie algebras (if time permits).
Text:
Introduction to Lie Algebras and Representation Theory, by J. E. Humphreys, Springer-Verlag, 3rd printing, 1980
Prerequisites:
Linear algebra (on the level of Math 350) and some acquaintance with abstract algebra (on the level of Math 551).
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Fall 2024
Anders Buch
Course Description:
Lie algebras are fundamental objects in algebra and important tools in mathematics and physics. The course will be an introduction to the theory of semisimple Lie algebras. The primary goal is the classification of semisimple Lie algebras in terms of root systems and Dynkin diagrams. The secondary goal (as time permits) is to classify the finite dimensional representations of a semisimple Lie algebra, and to prove Weyl's character formula.
Text:
Humphreys: Introduction to Lie Algebras and Representation Theory (GTM 9)
Prerequisites:
Linear algebra and Abstract Algebra 640:551
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Fall 2023
Pham Tiep
Course Description:
Semisimple Lie algebras are some of the fundamental concepts of algebra, which have proved to be useful in many areas of mathematics and physics. This course will be an introduction to the theory of semisimple Lie algebras, with emphasis on representations.
Topics will include elementary properties of Lie algebras, theorems of Lie and Cartan, conjugacy of Borel subalgebras, root systems and their construction, structure and classification of semisimple Lie algebras, and representation theory of semisimple Lie algebras (if time permits).
Text:
Introduction to Lie Algebras and Representation Theory, by J. E. Humphreys, Springer-Verlag, 3rd printing, 1980
Prerequisites:
Good knowledge of linear algebra and some acquaintance with abstract algebra (say first semester of Abstract Algebra for graduate students)
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Spring 2023
Siddhartha Sahi
Subtitle:
Lie Algebras
Course Description:
This will be a highly individualized course covering topics in Lie theory. It is suitable for students at several different levels. Here are some possibilities:
(1) Learn about Lie algebras and/or Lie groups as a possible minor/major topic for the qualifying exams
(2) Learn ideas from Lie theory and representation theory suitable for application to another area
(3) Work on an open problem in Lie theory and learn the relevant material along the way
Students can work on one or more topic, either individually or in teams. If you are interested in this course, please send me an e-mail, describing your background and interests.
Text:
None
Prerequisites:
Basic grad courses
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Fall 2021
Vladimir Retakh
Course Description:
This course will be an introduction to the theory of semisimple Lie algebras.
We will carry out the Cartan-Killing classification of semisimple Lie algebras, and the Cartan-Weyl classification of their irreducible finite-dimensional representations A lot of examples will be considered.
The only prerequisite is a good understanding of linear algebra. Thus the course should be accessible to first and second year graduate students, and even to advanced undergraduate students. This course will be of interest to students planning to work in Lie groups, differential geometry and topology, harmonic analysis, Lie algebras, algebra and algebraic geometry, mathematical and theoretical physics.
Text:
William Fulton, Joe Harris, "Representation Theory, A first course"
Prerequisites:
A standard course in Linear Algebra and a first semester of Abstract Algebra.
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Schedule of Sections: