November 25, 2019

Speaker: Tamar Lichter

Title: Quadratic Forms over Fields

Abstract: A quadratic form is a homogeneous polynomial of degree two. Quadratic forms are concrete and easy to write down, and they have many interesting features. For example, quadratic forms can be used to detect whether certain algebras are isomorphic; one can study the orthogonal group of a quadratic form; and there is a useful way to pass from quadratic forms over Q to quadratic forms over the p-adic numbers and R. We'll discuss some of these features of quadratic forms, and then delve into some deeper theory by studying the Witt ring of a field F: a ring whose elements are (almost) quadratic forms over F.

September 9, 2019

Speaker: Jason Saied

Title: Matrix Lie groups and their Lie algebras

Abstract: In this introductory talk, we will discuss Lie groups, groups with smooth manifold structures that make multiplication and inversion into smooth maps. To avoid the analytic complications of the general case, we will focus on matrix Lie groups. We will construct the Lie algebra of a matrix Lie group and prove some results allowing us to reduce questions about Lie groups (more complicated, analytic objects) to questions about their Lie algebras (finite dimensional vector spaces with an extra operation). I will only assume basic knowledge of groups, vector spaces, and undergraduate calculus.

September 23, 2019

Speaker: Tamar Lichter

Title: Introduction to Semisimple Lie Algebras

Abstract: This is the first in a series of three introductory talks on Lie algebras, with a focus on semisimple Lie algebras. A Lie algebra is a vector space with a particular (somewhat strange-looking) non-associative binary operation. These "algebras" may look odd at first, but it turns out that semisimple Lie algebras have quite a lot of structure and are subject to some surprisingly powerful theorems. Many of the really cool theorems about semisimple Lie algebras will be relegated to the next few GARTS talks - but in this talk we'll introduce the basic terminology and tools, and at least one big theorem.

Note: This talk is intended to be accessible to all graduate students, including first-year students. We'll only assume knowledge of some linear algebra and abstract algebra. In particular, to understand this talk, it's not necessary to have attended the previous GARTS talk.

September 30, 2019

Speaker: Johnny Fonseca

Title: Continuation of Semisimple Lie Algebras

Abstract: This talk will begin with an introduction to the abstract Jordan decomposition for semisimple Lie algebras, and (hopefully) it will end with very cool diagrams classifying them all. In particular, semisimple Lie algebras were shown to be decomposable as a direct sum of simple Lie algebras. As a consequence, to determine all possible semisimple Lie algebras, it is equivalent to classify all simple Lie algebras. This talk aims to introduce the main tools in obtaining this classification: root systems and Dynkin diagrams. As it turns out, classifying simple Lie algebras is equivalent to classifying irreducible root systems, which is further equivalent to classifying their corresponding Dynkin diagrams. To motivate their definitions, the structure of sl(2,F) and its representation theory will be examined; often called "sl2-theory." This theory will then be implemented in order to give semisimple Lie algebras a very nice decomposition relative to a certain set of weights - this is its root system.

October 7, 2019

Speaker: Songhao Zhu

Title: Classification and Representation Theory of Semisimple Lie Algebras (1/2)

Abstract: We will review the root space decomposition of semisimple Lie algebras and the notion of root system discussed by the last speaker with one or two motivating examples, and use Dynkin diagram to introduce the classification result of all simple Lie algebras. Weyl group will be defined without too many technicalities. We will also give a uniform way (Serre's Theorem) to construct semisimple Lie algebras with the help of universal enveloping algebra, which serves as a tool to study the representation theory of semisimple Lie algebras.

October 21, 2019

Speaker: Songhao Zhu

Title: Classification and Representation Theory of Semisimple Lie Algebras (2/2)

Abstract: We will continue the journey of the representation theory of semisimple Lie algebras. A brief review on root space decomposition, PBW theorem together with Serre's theorem will be given before we revisit the representation theory of sl2. With the sl2 theory in mind, we will explore the analogy for a general semisimple Lie algebra, and how to construct irreducible modules. The classification of them will be stated with the help of weight theory. Detailed description, namely, dimension and multiplicity of any such module will be introduced.

October 28, 2019

Speaker: Yael Davidov

Title: Generalizing the Quaternions

Abstract: In this talk we will take the quaternions, a well-known algebraic object famously discovered by Hamilton, and study a set of objects which generalize them called (confusingly) quaternion algebras. Very little prior knowledge is expected and this talk has nothing to with Lie algebras (to my knowledge). In the talk I will touch on some connections to quadratic forms, algebraic geometry, and number theory and we might prove some theorems.

November 11, 2019

Speaker: Terence Coelho

Title: Introduction to Affine Lie Algebras

Abstract: In the early talks of the semester, we’ve seen a natural correspondence between the finite-dimensional simple Lie algebras (over C), the indecomposable root systems, and the Cartan matrices. We’ve also seen how one can use this correspondence to classify the finite-dimensional simple Lie algebras and better understand their structure and representations. Infinite dimensional Lie algebras can be immensely complicated in general, but one can identify a “friendly” family of them by slightly relaxing the conditions on Cartan matrices and building Lie algebras from these. In this talk, I will go over the parallels between these “affine” Lie algebras and the finite-dimensional simple Lie algebras we’ve seen earlier. I will also discuss the general idea behind how to classify them and construct tangible realizations of these algebras.

November 18, 2019

Speaker: Sven Möller

Title: Introduction to Vertex Algebras

Abstract: I will give a gentle introduction to vertex (operator) algebras focusing on parallels with Lie algebras.