# Lie algebra

In mathematics, a Lie algebra (pronounced /liː/ LEE) is a vector space together with an operation called the Lie bracket, an alternating bilinear map , that satisfies the Jacobi identity. The Lie bracket of two vectors and is denoted . The vector space together with this operation is a non-associative algebra, meaning that the Lie bracket is not necessarily associative. Lie algebras are closely related to Lie groups, which are groups that are also smooth manifolds: any Lie group gives rise to a Lie algebra, which is its tangent space at the identity. Conversely, to any finite-dimensional Lie algebra over real or complex numbers, there is a corresponding connected Lie group unique up to finite coverings (Lie's third theorem). This correspondence allows one to study the structure and classification of Lie groups in terms of Lie algebras. In physics, Lie groups appear as symmetry groups of physical systems, and their Lie algebras (tangent vectors near the identity) may be thought of as infinitesimal symmetry motions. Thus Lie algebras and their representations are used extensively in physics, notably in quantum mechanics and particle physics. An elementary example is the space of three dimensional vectors with the bracket operation defined by the cross product This is skew-symmetric since , and instead of associativity it satisfies the Jacobi identity: This is the Lie algebra of the Lie group of rotations of space, and each vector may be pictured as an infinitesimal rotation around the axis , with velocity equal to the magnitude of . The Lie bracket is a measure of the non-commutativity between two rotations: since a rotation commutes with itself, we have the alternating property . (Wikipedia).

Lie groups: Lie algebras

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

Axioms of Lie algebra theory

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

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

Introduction to Lie algebras

In this clip I casually give a roundup of some of my current interests and also recommend you some literature. Get into Lie algebras, Lie groups and algebraic groups. Do it now! https://en.wikipedia.org/wiki/Lie_algebra http://www.jmilne.org/math/index.html

From playlist Algebra

Lie Groups and Lie Algebras: Lesson 18- Group Generators

Lie Groups and Lie Algebras: Lesson 18- Generators This is an important lecture! We work through the calculus of *group generators* and walk step-by-step through the exploitation of analyticity. That is, we use the Taylor expansion of the continuous functions associated with a Lie group o

From playlist Lie Groups and Lie Algebras

Lie Groups and Lie Algebras: Lesson 13 - Continuous Groups defined

Lie Groups and Lie Algebras: Lesson 13 - Continuous Groups defined In this lecture we define a "continuous groups" and show the connection between the algebraic properties of a group with topological properties. Please consider supporting this channel via Patreon: https://www.patreon.co

From playlist Lie Groups and Lie Algebras

Lie Groups and Lie Algebras: Lesson 16 - representations, connectedness, definition of Lie Group

Lie Groups and Lie Algebras: Lesson 16 - representations, connectedness, definition of Lie Group We cover a few concepts in this lecture: 1) we introduce the idea of a matrix representation using our super-simple example of a continuous group, 2) we discuss "connectedness" and explain tha

From playlist Lie Groups and Lie Algebras

Jean Michel BISMUT - Fokker-Planck Operators and the Center of the Enveloping Algebra

The heat equation method in index theory gives an explicit local formula for the index of a Dirac operator. Its Lagrangian counterpart involves supersymmetric path integrals. Similar methods can be developed to give a geometric formula for semi simple orbital integrals associated with the

From playlist Integrability, Anomalies and Quantum Field Theory

Lie Groups for Deep Learning w/ Graph Neural Networks

Lie Groups encode the symmetry of systems. We examine actions of a Lie group on a vector space, given their algebraic, topological and analysis based connectome. Deep Learning algorithms for Graph Neural Networks (GNN) are non trivial, and to understand them Lie Groups are essential! A r

Lie groups: Poincare-Birkhoff-Witt theorem

This lecture is part of an online graduate course on Lie groups. We state the Poincare-Birkhoff Witt theorem, which shows that the universal enveloping algebra (UEA) of a Lie algebra is the same size as a polynomial algebra. We prove it for Lie algebras of Lie groups and sketch a proof of

From playlist Lie groups

Is the variety of singular tuples of matrices a null cone? - Viswambhara Makam

Computer Science/Discrete Mathematics Seminar II Topic: Is the variety of singular tuples of matrices a null cone? - Speaker: Viswambhara Makam Affiliation: Member, School of Mathematics Date: February 25, 2020 For more video please visit http://video.ias.edu

From playlist Mathematics

Lie groups: Baker Campbell Hausdorff formula

This lecture is part of an online graduate course on Lie groups. We state the Baker Campbell Hausdorff formula for exp(A)exp(B). As applications we show that a Lie group is determined up to local isomorphism by its Lie algebra, and homomorphisms from a simply connected Lie group are deter

From playlist Lie groups

Lie Groups and Lie Algebras: Lesson 38 - Preparation for the concept of a Universal Covering Group

Lie Groups and Lie Algebras: Lesson 38 - Preparation for the Universal Covering Group concept In this lesson we examine another amazing connection between the algebraic properties of the Lie groups with topological properties. We will lay the foundation to understand how discrete invaria

From playlist Lie Groups and Lie Algebras

Lie Groups and Lie Algebras: Lesson 39 - The Universal Covering Group

Lie Groups and Lie Algebras: Lesson 39 - The Universal Covering Group We are finally in position to understand the nature of the Universal Covering Group and its connection to all the Lie groups which share a single Lie algebra. This is a critical lecture! In this lecture we simply state

From playlist Lie Groups and Lie Algebras

Lie groups: Introduction

This lecture is part of an online graduate course on Lie groups. We give an introductory survey of Lie groups theory by describing some examples of Lie groups in low dimensions. Some recommended books: Lie algebras and Lie groups by Serre (anything by Serre is well worth reading) Repre

From playlist Lie groups

Moduli of p-divisible groups (Lecture 4) by Ehud De Shalit

PROGRAM PERFECTOID SPACES ORGANIZERS: Debargha Banerjee, Denis Benois, Chitrabhanu Chaudhuri, and Narasimha Kumar Cheraku DATE & TIME: 09 September 2019 to 20 September 2019 VENUE: Madhava Lecture Hall, ICTS, Bangalore Scientific committee: Jacques Tilouine (University of Paris, France

From playlist Perfectoid Spaces 2019