Lie algebras | Representation theory
In mathematics, an affine Lie algebra is an infinite-dimensional Lie algebra that is constructed in a canonical fashion out of a finite-dimensional simple Lie algebra. Given an affine Lie algebra, one can also form the associated affine Kac-Moody algebra, as described below. From a purely mathematical point of view, affine Lie algebras are interesting because their representation theory, like representation theory of finite-dimensional semisimple Lie algebras, is much better understood than that of general Kac–Moody algebras. As observed by Victor Kac, the character formula for representations of affine Lie algebras implies certain combinatorial identities, the Macdonald identities. Affine Lie algebras play an important role in string theory and two-dimensional conformal field theory due to the way they are constructed: starting from a simple Lie algebra , one considers the loop algebra, , formed by the -valued functions on a circle (interpreted as the closed string) with pointwise commutator. The affine Lie algebra is obtained by adding one extra dimension to the loop algebra and modifying a commutator in a non-trivial way, which physicists call a quantum anomaly (in this case, the anomaly of the WZW model) and mathematicians a central extension. More generally, if σ is an automorphism of the simple Lie algebra associated to an automorphism of its Dynkin diagram, the twisted loop algebra consists of -valued functions f on the real line which satisfythe twisted periodicity condition f(x + 2π) = σ f(x). Their central extensions are precisely the twisted affine Lie algebras. The point of view of string theory helps to understand many deep properties of affine Lie algebras, such as the fact that the characters of their representations transform amongst themselves under the modular group. (Wikipedia).
In this video I write down the axioms of Lie algebras and then discuss the defining anti-symmetric bilinear map (the Lie bracket) which is zero on the diagonal and fulfills the Jacobi identity. I'm following the compact book "Introduction to Lie Algebras" by Erdmann and Wildon. https://gi
From playlist Algebra
This lecture is part of an online graduate course on Lie groups. We define the Lie algebra of a Lie group in two ways, and show that it satisfied the Jacobi identity. The we calculate the Lie algebras of a few Lie groups. For the other lectures in the course see https://www.youtube.co
From playlist Lie groups
The Lie-algebra of Quaternion algebras and their Lie-subalgebras
In this video we discuss the Lie-algebras of general quaternion algebras over general fields, especially as the Lie-algebra is naturally given for 2x2 representations. The video follows a longer video I previously did on quaternions, but this time I focus on the Lie-algebra operation. I st
From playlist Algebra
algebraic geometry 26 Affine algebraic sets and commutative rings
This lecture is part of an online algebraic geometry course, based on chapter I of "Algebraic geometry" by Hartshorne. It covers the relation between morphisms of affine algebraic sets and homomorphisms of commutative rings. As examples it describes some homomorphisms of commutative rings
From playlist Algebraic geometry I: Varieties
Commutative algebra 19 Affine schemes
This lecture is part of an online course on commutative algebra, following the book "Commutative algebra with a view toward algebraic geometry" by David Eisenbud. In this lecture we give an introduction to the affine scheme of a ring, that is heavily used in algebraic geometry. We check t
From playlist Commutative algebra
The Weyl algebra and the Heisenberg Lie algebra
In this video we give a simple teaser into the world of operator algebras. In particular, we talk about the Weyl algebra and compute some expressions that fulfill the property which defines the Heisenberg Lie algebra http://math.uchicago.edu/~may/REU2012/REUPapers/Lingle.pdf https://en.w
From playlist Algebra
Lie groups: Lie groups and Lie algebras
This lecture is part of an online graduate course on Lie groups. We discuss the relation between Lie groups and Lie algebras, and give several examples showing how they behave differently. Lie algebras turn out to correspond more closely to the simply connected Lie groups. We then explain
From playlist Lie groups
Differential Isomorphism and Equivalence of Algebraic Varieties Board at 49:35 Sum_i=1^N 2/(x-phi_i(y,t))^2
From playlist Fall 2017
Martina Lanini, Introduction - 9 December 2014
Minicourses of the session "Vertex algebras, W-algebras, and applications" (2014) http://www.crm.sns.it/event/321/speakers.html?page=1#title INdAM Intensive research period Perspectives in Lie Theory Session 1: Vertex algebras, W-algebras, and applications Mini-courses Tomoyuki Arakawa
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Representations of affine Lie algebras vs quantum groups: fusion, factorization and E_2...- Lin Chen
Short Talks by Postdoctoral Members Topic: Representations of affine Lie algebras vs quantum groups: fusion, factorization and E_2 structures Speaker: Lin Chen Affiliation: Member, School of Mathematics Date: September 22, 2021
From playlist Mathematics
Joel Kamnitzer: Categorical g-actions for modules over truncated shifted Yangians
CIRM VIRTUAL CONFERENCE Given a representation V of a reductive group G, Braverman-Finkelberg-Nakajima defined a Poisson variety called the Coulomb branch, using a convolution algebra construction. This variety comes with a natural deformation quantization, called a Coulomb branch algebr
From playlist Virtual Conference
Holomorphic Cartan geometries on simply connected manifolds by Sorin Dumitrescu
Discussion Meeting Complex Algebraic Geometry ORGANIZERS: Indranil Biswas, Mahan Mj and A. J. Parameswaran DATE:01 October 2018 to 06 October 2018 VENUE: Madhava Lecture Hall, ICTS, Bangalore The discussion meeting on Complex Algebraic Geometry will be centered around the "Infosys-ICT
From playlist Complex Algebraic Geometry 2018
Shrawan Kumar: Root components for tensor product of affine Kac-Moody Lie algebra modules
SMRI Algebra and Geometry Online: Shrawan Kumar (University of North Carolina) Abstract: This is a joint work with Samuel Jeralds. Let 𝔤 be an affine Kac-Moody Lie algebra and let λ, µ be two dominant integral weights for 𝔤. We prove that under some mild restriction, for any positive root
From playlist SMRI Algebra and Geometry Online
Points and Lines in the Affine Plane | Algebraic Calculus One | Wild Egg
This is the first lecture in the Algebraic Calculus One course, which will present an exciting new approach to calculus, sticking with rational numbers and high school algebra. The course will be carefully framed on careful definitions, explicit examples and concrete computations. In thi
From playlist Algebraic Calculus One
Lie groups: Positive characteristic is weird
This lecture is part of an online graduate course on Lie groups. We give several examples to show that, over fields of positive characteristic, Lie algebras can behave strangely, and have a weaker connection to Lie groups. In particular the Lie algebra does not generate the ring of all in
From playlist Lie groups
Anthony Henderson: Hilbert Schemes Lecture 1
SMRI Seminar Series: 'Hilbert Schemes' Lecture 1 Introduction Anthony Henderson (University of Sydney) This series of lectures aims to present parts of Nakajima’s book `Lectures on Hilbert schemes of points on surfaces’ in a way that is accessible to PhD students interested in representa
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Geordie Williamson: Langlands and Bezrukavnikov II Lecture 9
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From playlist Geordie Williamson: Langlands correspondence and Bezrukavnikov’s equivalence
This lecture is part of an online graduate course on Lie groups. This lecture is about Lie's theorem, which implies that a complex solvable Lie algebra is isomorphic to a subalgebra of the upper triangular matrices. . For the other lectures in the course see https://www.youtube.com/playl
From playlist Lie groups
algebraic geometry 17 Affine and projective varieties
This lecture is part of an online algebraic geometry course, based on chapter I of "Algebraic geometry" by Hartshorne. It covers the relation between affine and projective varieties, with some examples such as a cubic curve and the twisted cubic.
From playlist Algebraic geometry I: Varieties
Representation theory and geometry – Geordie Williamson – ICM2018
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From playlist Plenary Lectures