MATH7301 Lie Groups and Lie Algebras (2026 spring)
Lecture 1: Manifolds.
We review the basic notions of smooth manifolds and the properties we will need later, such as tangent spaces, regular values, the inverse function theorem, and so on.
Lecture 2: Lie groups I.
We introduce the category of Lie groups. We study the connected component, Lie subgroups and covering groups.
Lecture 3: Lie groups II.
We study basic notions and properties of homogeneous spaces, Lie group actions and representations.
Lecture 4: Lie algebra of Lie group.
We define the Lie algebra of a Lie group as the space of Left invatiant vector fields on the Lie group, and discuss the category of Lie algebras. We introduce the exponential map, and use it to study the relationship between Lie groups and Lie algebras.
Lecture 5: Fundamental theorems of Lie theory.
We state three fundamental theorems of Lie theory and prove the first two.
Lecture 6: Representation of compact Lie group.
We introduce the category of representations of compact Lie groups. We study the matrix coefficients and character theory.
Lecture 7: Peter-Weyl Theorem.
We state the Peter-Weyl Theorem, and prove that the natural map occurring in the theorem is injective, which is a corollary of the properties we studied in the previous class.