Sept. 15, 2023
Speaker: Madison Crim
Title: Introduction to Representation Theory of Lie Algebras
Abstract: I will provide background on what a Lie algebra is including some examples. We’ll discuss semisimple Lie algebras which give rise to important results of finite dimensional representations such as Weyl’s theorem on complete reducibility. These tools will then allow us to examine representations of the simple Lie algebra sl(2, F).
Sept. 22, 2023
Speaker: Ching Hsien Lee
Title: Root Systems
Abstract: The classification of simple Lie algebras arises from their correspondence with root systems. The root systems consist of finite sets of vectors in Euclidean space, possessing distinctive symmetry and integral characteristics. We will delve into this connection and elucidate how it impacts the structure of simple Lie algebras and their associated modules.
Sept. 29, 2023
Speaker: Sumit Singh
Title: Freudenthal's Multiplicity formulas
Abstract: The Problem addressed in this talk is about finding the dimension of Weight Spaces of Irreducible representation of finite dimensional Lie Algebra corresponding to a dominant weight. This can be solved recursively using Freudenthal Multiplicity Formula. The main idea in the proof is to calculate the action of Universal Casimir Element, which acts as a scalar on the space.
Oct. 6, 2023
Speaker: Dan Tan
Title: PBW Theorem
Abstract: Universal enveloping algebras relate the representation theory of Lie algebras to the representation theory of associative unital algebras. The PBW Theorem provides a basis for a given universal enveloping algebra. Continuing our series of introductory talks on Lie algebra, we’ll state and prove the PBW Theorem as presented in Humphreys.
Oct. 13, 2023
Speaker: Jishen Du
Title: Integrable Highest Weight Modules, and Weyl-Kac Character Formula
Abstract: Given a generalized Cartan matrix(GCM) A, one can construct the so-called Kac-Moody algebra g(A), which is roughly speaking an infinite dimensional Lie algebra with a nice triangular decomposition (also called Cartan decomposition). I will talk about some representation theory of g(A), define integrable highest weight module, and finally prove the famous Weyl-Kac character formula. If time permits, I will talk about some application of it.
Oct. 27, 2023
Speaker: Dennis Hou
Title: The Infinite Symmetric Group
Nov. 3 & 10, 2023
Speaker: Hong Chen
Title: Symmetric Functions and Characters of Symmetric Groups
Abstract: I will first give an introduction to the algebra of symmetric functions, including three sets of free generators $e_n$, $h_n$ and $p_n$ (elementary, complete homogeneous and power sum), two bases $m_\lambda$ and $s_\lambda$ (monomial and Schur), many relations among them, and a scalar product. Then I will talk about the relation between symmetric functions and characters of symmetric groups, in particular, I will show that the algebra of symmetric functions is isometrically isomorphic to the algebra of characters of symmetric groups. I will also mention the hook-length formula.
Dec. 1, 2023
Speaker: Nick Backes
Title: Representations of SL(2,R) and sl(2,R)
Fall 2022 Talks (Songhao)
Spring 2021 Talks (Jason Saied)