In the mathematical field of Lie theory, there are of a compact Lie algebra. Extrinsically and topologically, a compact Lie algebra is the Lie algebra of a compact Lie group; this definition includes tori. Intrinsically and algebraically, a compact Lie algebra is a real Lie algebra whose Killing form is negative definite; this definition is more restrictive and excludes tori,. A compact Lie algebra can be seen as the smallest real form of a corresponding complex Lie algebra, namely the complexification. (Wikipedia).
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
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
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
Lie Groups and Lie Algebras: Lesson 32: Parameters Space and Compactness
Lie Groups and Lie Algebras: Lesson 32: Parameters Space and Compactness I this lecture we prepare ourselves for the study of the homology of SO(3) and SU(2). Homology will be our way of beginning to understand the difference between these groups. This lecture ends abruptly, but it was
From playlist Lie Groups and Lie Algebras
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
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 34 -Introduction to Homotopy
Lie Groups and Lie Algebras: Introduction to Homotopy In order to proceed with Gilmore's study of Lie groups and Lie algebras we now need a concept from algebraic topology. That concept is the notion of homotopy and the Fundamental Group of a topological space. In this lecture we provide
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
Lie Groups and Lie Algebras: Lesson 25 - the commutator and the Lie Algebra
Lie Groups and Lie Algebras: Lesson 25 - the commutator In this lecture we discover how to represent an infinitesimal commutator of the Lie group using a member of the Lie algebra. We promote the vector space spawned by the group generators to an algebra. Please consider supporting this
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
Representations of p-adic groupsz - Jessica Fintzen
Workshop on Representation Theory and Analysis on Locally Symmetric Spaces Topic: Representations of p-adic groupsz Speaker: Jessica Fintzen Affiliation: University of Michigan; Member, School of Mathematics Date: March 5, 2018 For more videos, please visit http://video.ias.edu
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Higgs bundles and higher Teichmüller components (Lecture 1) by Oscar Garcia
DISCUSSION MEETING : MODULI OF BUNDLES AND RELATED STRUCTURES ORGANIZERS : Rukmini Dey and Pranav Pandit DATE : 10 February 2020 to 14 February 2020 VENUE : Ramanujan Lecture Hall, ICTS, Bangalore Background: At its core, much of mathematics is concerned with the problem of classif
From playlist Moduli Of Bundles And Related Structures 2020
Higgs bundles and higher Teichmüller components (Lecture 2) by Oscar García-Prada
DISCUSSION MEETING : MODULI OF BUNDLES AND RELATED STRUCTURES ORGANIZERS : Rukmini Dey and Pranav Pandit DATE : 10 February 2020 to 14 February 2020 VENUE : Ramanujan Lecture Hall, ICTS, Bangalore Background: At its core, much of mathematics is concerned with the problem of classif
From playlist Moduli Of Bundles And Related Structures 2020
David Zywina, Computing Sato-Tate and monodromy groups.
VaNTAGe seminar on May 5, 2020. License: CC-BY-NC-SA Closed captions provided by Jun Bo Lau.
From playlist The Sato-Tate conjecture for abelian varieties
Lie Groups and Lie Algebras: Lesson 41: Elementary Representation Theory I
Lie Groups and Lie Algebras: Lesson 41: Elementary Representation Theory I I wanted to begin a more intricate example of the principle of a Universal Covering group, but I think I need to cover a little background material. We need to get a grip on what is meant by "Representation Theory"
From playlist Lie Groups and Lie Algebras
The Embedding Problem of Infinitely Divisible Probability Measures on Groups by Riddhi Shah
PROGRAM : ERGODIC THEORY AND DYNAMICAL SYSTEMS (HYBRID) ORGANIZERS : C. S. Aravinda (TIFR-CAM, Bengaluru), Anish Ghosh (TIFR, Mumbai) and Riddhi Shah (JNU, New Delhi) DATE : 05 December 2022 to 16 December 2022 VENUE : Ramanujan Lecture Hall and Online The programme will have an emphasis
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Colloquio De Giorgi - Christopher Deninger - Primes, knots and periodic orbits - 11 marzo 2022
In the 1960s Manin and Mazur noted that from the viewpoint of étale topology there was an intriguing analogy between prime numbers embedded into the spectrum of the integers and knots in 3-space. Later Kapranov, Reznikov, Morishita and other authors discovered further intriguing analogies
From playlist Talks of Mathematics Münster's reseachers
Askold Khovanskii: Complex torus, its good compactifications and the ring of conditions
Abstract: Let X be an algebraic subvariety in (ℂ∗)n. According to the good compactifification theorem there is a complete toric variety M⊃(ℂ∗)n such that the closure of X in M does not intersect orbits in M of codimension bigger than dimℂX. All proofs of this theorem I met in literature ar
From playlist Algebraic and Complex Geometry
Lie Groups and Lie Algebras: Lesson 31 - U(2,C) and GL(1,Q)
Lie Groups and Lie Algebras: Lesson 31 - U(2,C) and GL(1,Q) In this lecture we back up and deploy the basis elements we eliminated in the su(2) and so(3) algebras when we enforced the determinants to be equal to 1. This expands the algebras to u(2) and o(3) and generates the groups U(2) a
From playlist Lie Groups and Lie Algebras