18.748 Topics in Lie Theory
This course is an introduction to Yangians and their representation theory. Yangian of a simple Lie algebra g is a Hopf algebra deformation of the universal enveloping of the current algebra g[t], historically one of the first examples of quantum groups. Such Hopf algebras naturally arise in many different areas, from the Representation Theory of classical Lie algebras to Quantum Integrable Systems, Algebraic and Symplectic Geometry (in particular, in the phenomenon of symplectic duality), and Algebraic Combinatorics. We will start with some motivating examples coming from classical Representation Theory (Gelfand-Tsetlin bases and Olshansky centralizer construction), then develop the general theory of Yangians and their representations, and finally discuss some applications to Integrable Systems, Geometry, and Combinatorics.
For the first half of the course, the standard source is the book ``Yangians and Classical Lie Algebras'' by Alexander Molev. For the second half of the course, I will provide some references during the semester.
To officially pass the course, it will be required to solve the homework assignments, there will be 3 of them during the semester.
Prerequisites: Lie Groups and Lie Algebras 18.745/18.755
Lecture notes: 2/5/2024 Yangians Lecture 1.pdf
2/7/2024 Yangians Lecture 2.pdf
2/12/2024 Yangians Lecture 3.pdf
2/14/2024 Yangians Lecture 4.pdf
2/20/2024 Yangians Lecture 5.pdf
2/21/2024 and 2/26/2024 Yangians Lecture 6.pdf
2/28/2024 Yangians Lecture 7.pdf
3/4/2024 and 3/6/2024 Yangians Lecture 8.pdf
3/11/2024 and 3/13/2024 Yangians Lecture 9.pdf
3/18/2024 Yangians Lecture 10.pdf
3/20/2024 and 4/1/2024 Yangians Lecture 11.pdf
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Exercises: Set #1 due 3/22/2024 Exercises_Math_18_748_Set_1.pdf
Exercises: Set #2 due 5/13/2024 Exercises_Math_18_748_Set_2-1.pdf
Course Summary:
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